For generalized Nash equilibrium problems (GNEP) with shared constraints we focus on the notion of normalized Nash equilibrium in the nonconvex setting. The property of nondegeneracy for normalized Nash equilibria is introduced. Nondegeneracy refers to GNEP-tailored versions of linear independence constraint qualification, strict complementarity and second-order regularity. Surprisingly enough, nondegeneracy of normalized Nash equilibrium does not prevent from degeneracies at the individual players' level. We show that generically all normalized Nash equilibria are nondegenerate. Moreover, nondegeneracy turns out to be a sufficient condition for the local uniqueness of normalized Nash equilibria. We emphasize that even in the convex setting the proposed notion of nondegeneracy differs from the sufficient condition for (global) uniqueness of normalized Nash equilibria, which is known from the literature.