In $R^3$, since the spheres are concentric, not only all faces are regular, but also all edges are of the same length, and all faces are inscribed in circles of the same radius, hence are congruent. Also, all dihedral angles between faces with a common edge are equal, which implies that all vertices are of the same valence.  This makes the polytope regular. It seems that this reasoning can be generalized to all dimensions.