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.
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
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.
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.
