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?)