Why does the $L_2$ norm give the shortest path between 2 points? Why not the  $L_1$ or  $L_3$ distances? Is there some deep reason why the universe (at least at human scales) looks pretty much Euclidean? 
Could we imagine a different universe where a different  $L_p$ metric would seem "natural"?
I know it's kind of a deep question, but the specialness of 2 here has always made me wonder. 
 A: If we want a sense of localness (or calculus to work), we'd like to be able to obtain the length by adding up the length from pieces of the path (for example using a ruler, or counting paces as we walk along the path between two points).
However, even considering just two dimensions we see something interesting for $L_p$.
$$\left(|x|^p + |y|^p\right)^{1/p} = \sum_{i=1}^N \left(\left|\frac{x}{N}\right|^p + \left|\frac{y}{N}\right|^p\right)^{1/p}$$
This trivially works with $p=1$, and due to a special symmetry at $p=2$ it works there as well.  This will not work for other $p\neq 0$ (I am unsure of how to extend the definition to check $p=0$). 
The special symmetry at $p=2$ is that the distance measurement becomes rotationally invariant.  So the seemingly mundane reasons of 


*

*space has more than one dimension

*locality

*uniformity


seem to already select $L_2$ as special.  Any other choice would give a preferred coordinate system, and possibly break locality.
So what would a different universe in which $L_1$ or something else is the natural choice?  If you imagined an N dimensional Cartesian lattice world, so one with discrete lengths, and a clearly preferred coordinate basis, this would make $L_1$ a more natural choice.
I'm not sure of a good picture for a universe in which $L_p, p>2$ would be a natural choice. There would be preferred directions, and you could only consider an object as a whole (not in parts), which seems to suggest in such a hypothetical universe you couldn't even experience your life as a sequence of moments (which I guess would make sense if we have highly non-local physics and therefore causality is out the window).
Interesting question.
A: Comments to the question (v2):


*

*One nice property of the 2-norm (as compared to other norms, such as, e.g. the $p$-norm) is that it gives rise to an inner product via a so-called polarization trick. E.g. in the real case the polarization formula has 2 terms:
$$ \langle u, v \rangle ~:=~\frac{1}{4} || u+v ||^2 -\frac{1}{4} || u-v ||^2 .$$
There is a similar 4-term polarization formula in the complex case. See also this related Phys.SE post.

*It seems appropriate to mention that there exist metric theories, which are
not based on Riemannian manifolds with a metric
tensor and its corresponding
2-norm. One class of such generalized metric theories is Finsler
geometry, see e.g. arXiv.org.
