Is position fundamentally different from other observables? I apologize in advance for what might be a very naive question, and for its science-fictionesque flavor.  It's still, I think, a real physics question.
Suppose I have a (quantum) particle, whose state is represented by some element $x$ of some Hilbert space $H$.  Given an observable $O$, I can express $x$ as a sum of eigenstates of $O$.   For example, I could take $O$ to represent the particle's position --- but so far, at least, "position" --- as opposed to any other observable --- seems to play no special role.
On the other hand, at the macro level, our brains seem to assign a very special role to position --- namely, when we're deciding whether to think of some collection of particles as an "object", one of our main criteria is (I think) whether these particles occupy nearby positions, and whether they continue to do so over time.  Moreover, we ourselves are objects by this criterion.
One can (perhaps barely) imagine some other sort of creature that perceives a collection of particles as an object when (and only when) those particles have, say, similar momenta (regardless of their positions) and continue to do so over time --- or similar values of some linear combination of position and momentum, or similar values of some other observable $O$ for which we don't even have a name.  Presumably these creatures would themselves be collections of particles that are $O$-localized but not necessarily position-localized. 
But for us humans, position seems to play this unique role.  
Question 1.  Is the unique role of position a matter of physics or of biology?  In other words, is there anything in fundamental physics that rules out (or renders unlikely) the sort of $O$-localized (but not position-localized) creatures I'm imagining?  
Question 1A.  Let me reword that.  It seems clear that (spacetime) position plays a special role in relativity, which quite plausibly rules out my $O$-localized creatures.  So let me ask more specifically whether there is anything in quantum mechanics (as opposed to all of physics) that explains these creatures' absence.
Question 2.  If the answer to Question 1A is yes, then how does one reconcile this with the fact that any basis of the Hilbert Space $H$ can be transformed via an isomorphism to any other basis, which seems to say, in essence, that there can be nothing special about the position basis, and hence nothing special about the position operator?
Question 3. If relativity grants a special role to position and quantum mechanics does not, does that all by itself constitute a fundamental incompatibility between relativity and quantum mechanics?  And can it be viewed as a significant failure of quantum mechanics that it cannot account for this aspect of the world?
Edited to add:  I'm glad for the answers I've gotten so far, but I infer from some of them that I need to be clearer about what I'm asking.  The question is not just "Why can't there be $O$-creatures?".  Instead, it is:  "If we have good reasons for thinking that $O$-creatures are unlikely, while knowing that position-creatures like us exist, how can we reconcile this with the fact that quantum mechanics sees no essential difference between $O$ and position?  (Or am I wrong about the "no essential difference" part?)
 A: Short answer: the position-momentum symmetry of the Hamiltonian formalism (classical or quantum) is explicitly broken by the actual Hamiltonian in all but a few special cases. In particular, it is broken in any interacting theory because things interact when they have the same position, not when they have the same momentum.
A: First of all, perhaps better than saying that special relativity gives a special role to position (or spacetime), one can say that special relativity is the theory that is designed to describe spacetime, by providing the right (Lorentzian) metric for it. Compatibility of quantum physics and special relativity then boils down to compatibility of quantum physics with Lorentzian spacetimes, which is hopefully less of a shock or mystery. The quantum mechanical description of systems living on Lorentzian spacetimes is known to be fully consistent and goes by the name of quantum field theory. The consistency of quantum physics and special relativity is really as simple as that.
Secondly, as you correctly mentioned, position does not play any singular role in the conventional framework of quantum mechanics (in fact it is somewhat of a bad observable because of normalizability issues of its eigenvectors and so on). This could be the end of an answer to your question. However, to exhaust my knowledge, let me add that when it comes to the controversial issue of measurement, sometimes position basis acquires a unique role. The following example form a review paper on quantum decoherence illustrates this:

System-environment interaction Hamiltonians frequently describe a scattering process of surrounding particles (photons, air molecules, etc.) interacting with the system under study. Since the force laws describing such processes typically depend on some power of distance [...], the interaction Hamiltonian will usually commute with the position basis, such that, [...] the preferred basis will be in position space. The fact that position is frequently the determinate property of our experience can then be explained by referring to the dependence of most interactions on distance.

http://arxiv.org/abs/quant-ph/0312059
A: Q1. I doubt there would be this sort of $O$-localized creatures. Here is the reason.
First, your $O$-creature still has to be made up of particles. Since these particles are $O$-localized, it is reasonable to assume that your $O$-creature will spread over the whole space. Otherwise, your $O$-creature would be space-localized and become a trivial object that we know.
Well, we can further conclude that the interactions between the constituent particles are very weak. The long range interactions are gravity and coulomb interaction. Very likely is that your $O$-creature are neutral, which means the particles can influence each other only through the weakest gravity. Even though your $O$-creature is charged, the interaction forces still decay ~ $r^{-2}$ and the mean inter-particle spacing is $r\rightarrow +\infty$. Also, there would be no entanglement among the particles due to decoherence.
Since the inter-particle interaction is very weak (perhaps no interaction would be more appropriate), this creature is so "loose" that it's very unlikely for it to develop any complex structures, let alone consciousness. Even worse, your $O$-object may not be called an object at all, because the constituent particles evolve solely on their own without even "knowing" the existence of others.
Perhaps a more interesting question is "what would happen if an object is not so localized in space". Take the example of BEC of atoms. All atoms occupy the same state and their positions are smoothed in a localized volume that could be orders of magnitude larger than an ordinary atom.
A: This answer is from an experimental physicist, you know, the ones who call a spade a spade.

Question 1. Is the unique role of position a matter of physics or of biology? In other words, is there anything in fundamental physics that rules out (or renders unlikely) the sort of O-localized (but not position-localized) creatures I'm imagining?

Position does not hold  unique place in physics, we have fourier transorms that take our calculations from space and time to momentum and energy coordinates. The latter we observe with instruments and finally our eyes and brain.
It is biology in the end, because everything ends up in our brain/memories. We observe and map in our brains a three dimensional and time coordinate system to evolve a life.
Unlikely is something that needs calculation of probabilities for a stable system, which biology/life as we know it has developed in 3+1 dimensions by trial and error. Imagination is the limit for possible, but likelihood needs calculations.
To start with one would have to imagine the equivalent of chemistry and biology in the O coordinates, and how consciousness could change its coordinates. In 3+1 we use energy for our existence. There would have to be an atomic/chemistry/biology setup in momentum energy space that could mimic life as we know it. I is not a simple problem for the imagination. And then one has to add gravity to the mix.

Question 1A. Let me reword that. It seems clear that (spacetime) position plays a special role in relativity, which quite plausibly rules out my O-localized creatures. So let me ask more specifically whether there is anything in quantum mechanics (as opposed to all of physics) that explains these creatures' absence.

Quantum mechanics has been developed to model the world we find ourselves embedded in. It would possibly describe other creatures perceptions, assuming they have consciousness. It is a science fiction question though. If your O creatures were within our spacetime and energy/momentum range  I can see no reason QM would not model them.

Question 2. If the answer to Question 1A is yes, then how does one reconcile this with the fact that any basis of the Hilbert Space H can be transformed via an isomorphism to any other basis, which seems to say, in essence, that there can be nothing special about the position basis, and hence nothing special about the position operator?

I have answered this. It is biology that gives specialness to position versus momenta

Question 3. If relativity grants a special role to position and quantum mechanics does not, does that all by itself constitute a fundamental incompatibility between relativity and quantum mechanics? And can it be viewed as a significant failure of quantum mechanics that it cannot account for this aspect of the world?

There exists relativistic quantum mechanics after all. The Dirac equation. No incompatibility. There is no special role to position from relativity with respect to energy momentum. It is biology, dependent on chemistry etc that gives the special role to spacetime.
In my opinion of course. For a science fiction project it would be easier to use  the extra dimensions postulated in string theories, which also are different than our 3+1 , to create new creatures.
A: As far as we know, life happens naturally in systems that are complex and stable enough to support a mechanism of adaptive evolution over a sufficient time scale. Possibly, some other order parameter could take the place of time.

So let me ask more specifically whether there is anything in quantum mechanics (as opposed to all of physics) that explains these creatures' absence.

It might just be lack of imagination on our side or a sufficient difference in scale.
Heck, some superorganisms might be complex enough (an ant hill? a city? human society as a whole? the biosphere as a whole?) for consciousness to emerge, which doesn't mean we'll ever be able to prove this conclusively and say 'hello'. Add to that a completely different mode of perceiving reality without a notion of space-time, and we're even more out of our depth.
Note that evolution could even work if $O$-space weren't really as 'real' as position space - which, as far as I'm concerned, actually is the case for momentum:
The apparent duality between configuration space coordinates and conjugate momenta is just an artifact of our phase space formalisms and not necessarily an inherent feature of reality.
Take relativistic mechanics: It allows a (constrained) Hamiltonian formulation where momentum coordinates are first class. However, due to reparametrization invariance, another way to look at it is as worldline dynamics via jets of submanifolds, which is arguably a more fundamental description that lacks this duality.
A: This is a great and very deep question. You are completely correct that in non-relativistic quantum mechanics, position and momentum are on completely symmetric footing. There is no reason to prefer either basis over the other, or indeed to assume that "natural systems" (like people!) would have wavefunctions that are localized in one basis but not the other.
That having been said, in the real world, which is not described completely by non-relativistic QM, there is definitely a physical asymmetry between the two. All known fundamental interactions can be expressed in a form that is spatially local. (Although in the case of relativistic gauge QFTs, this "locality" is a subtle and slippery concept.) At larger scales like in condensed matter physics, we might expect some kind of effective non-locality to emerge in certain regimes from fundamentally local physics via many body effects, but in practice we find that this basically never happens. So the special status that we humans psychologically place on the position basis almost certainly has its origin in fundamental physics, not in biology. The locality of the macroscopic phenomena that we observe in our everyday lives almost certainly just "bubbles up" from the locality of the Standard Model and general relativity.
Of course, this just shifts the question to why the interactions in the fundamental physics models are observed to be spatially local in a way that breaks the symmetry between position and momentum. This is partially a philosophical question. In my opinion, it just originates from relativity, even in apparently nonrelativistic situations, and relativity is just the way things are and probably can't be explained at a deeper level, but others might disagree. (Relativity is certainly incompatible with non-relativisitic quantum mechanics, but these days most people probably define "quantum mechanics" broadly enough to include relativistic QFT.) So yes, in my opinion the special status of position in everyday life is fundamentally relativistic in origin.
Here's an amusing thought experiment to conclude. One could certainly imagine a quantum-mechanical Lagrangian or Hamiltonian consisting of a sum of interaction terms that are local in position space (as we actually find in the real world) and interactions terms that are local in momentum space. Such a model would be incompatible with relativity but logically self-consistent. It would produce physics very different from what we see in our world, and if intelligent beings could exist in such a universe then they probably wouldn't see the position basis as special. But we could also imagine a Lagrangian or Hamiltonian consisting only of interactions that are local in momentum space. Such a model might initially seem equally bizarre, but in fact it would produce physics identical to our own! We would just naturally work/think in the momentum basis and call the weird nonlocal basis the "position basis", and indeed we ourselves would be in (approximate) momentum eigenstates, but it would just be a change of names. So the real mystery of locality isn't "Why are all are observed interactions local in the position basis but not the momentum basis? What makes the position basis more special than the momentum basis?" It's "Why are all observed interactions local in the same one of the two conjugate bases, with no intermixing?" Once you accept that is the case, it's clearly natural to prefer the one basis in which they're all local, and think of that one as "real space".
A: Given a classical Lagrangian $\mathcal L(q^j(t), \dot q^j(t))$, for particles of fields, a quantization means considering the $q_j$ as operators $Q_j$, with non vanishing commutators with the operators corresponding to the canonical momenta $p_j(t) = \frac{\partial \mathcal L}{\partial \dot q^j(t)}$, that is $[Q^j(t), P_k(t')]_{|t=t'} \sim i\delta^j_k~ $, where $i$ is the imaginary unit.
In Quantum mechanics (particles), the variables $q^i(t)$  correspond to the position $x^i(t)$, and usually, the canonical momenta corresponds to the usual momenta $p^i(t)$ (if there is only the kinetic term $\frac{m}{2}\dot x^2$ in the Lagrangian). So the operator position $X^i(t)$, in Quantum mechanics, play a very special role, that no other observable can play. With only the kinetic term in the Lagrangian, we have $P_i(t) = m \dot X^i(t)$, so we have the Heisenberg constraints : $[X^j(t), P_k(t')]_{|t=t'}= i \hbar \delta^j_k$
