I think of the principle of superposition as a consequence of the vector sum. I think this principle should hold for any other form of Coulomb's Law. But in the book Introduction to Electrodynamics by David J. Griffiths, he writes that this is not a logical necessity, but an experimental fact. Please explain more elaborately.
I would say that the mathematical reason this holds is that Maxwell's equations are linear, and the usual boundary conditions we consider (that the field is zero infinitely far from sources) are homogeneous. In general, linear differential equations with homogeneous boundary conditions satisfy a "superposition principle" which says that the sum of two solutions is a solution.
Physically, the linearity of Maxwell's equations is tied into the fact that photons do not interact with one another. For non-abelian Yang Mills theories, the classical equations of motion are non-linear. On the quantum level, these non-linearities lead to interactions between the gauge bosons in the theory.
I think what the statement about "logical necessity" is trying to say is that people could have written down a non-linear theory which wouldn't have a superposition principle (e.g. classical non-abelian Yang Mills). However, it was the linear theory which agreed with observations of how the electric field behaved.
