The special properties extend into statistics, where you have the Gaussian distribution which feels both magical and universal, and is precisely the exponential of a (squared) Euclidean distance, i.e. exp(-(x - x0)^2).

I have the same feeling, that Cartesian coordinates and Euclidian distances are inherently connected as a natural pairing that is uniquely suited for producing the familiar reality that we inhabit and experience.

In my opinion it holds the same place in mathematics that water holds in biology and chemistry.

Cartesian coordinates and Euclidean distances are both great ideas, for different (orthogonal) reasons.

Cartesian coordinates are orthogonal, which is great.

Euclidean distance is great because it makes space flat and rotationally symmetric.