Comment on the above comment. See now that I completely misunderstood the question being asked.

A simple and natural characterisation was already used by Clark Maxwell to deduce what is now known as the Maxwell-Boltzmann formula for the distribution of molecular velocities in a gas. Its density is the only function of two variables which depends only on the distance from the origin and is representable as a product of two functions of one varable. (we give the two-dimensional version for simplicity). Maxwell solved the corresponding functional equation by inspection but it is easy to give a rigorous proof that the only solution (up to constants) is the Gaussian kernel $\exp (x^2+y^2)$.

Yes. The interesting question between those two extremes is the case of Lipschitz equivalence (for Banach spaces). Again much work has been done on this case, e.g. by Lindenstrauss et al.