Comment by shiandow

Most of the special properties can be traced back to its special relationship with the inner product. And inner products have somewhat more elementary properties, so in that sense it explains the special position of the euclidean norm.

This has nothing to do with the coordinates by the way. If you want a different norm you'll first have to figure out an alternative to the bilinearity that gives the inner product its special properties.

Though bilinearity is pretty special itself, given the link between the tensor space and the linear algebra equivalent of currying.