We study the multiexciton optical spectrum of vertically coupled GaN/AlN quantum dots with a realistic three-dimensional direct-diagonalization approach for the description of few-particle Coulomb-correlated states. We present a detailed analysis of the fundamental properties of few- particle/exciton interactions peculiar of nitride materials. The giant intrinsic electric fields and the high electron/hole effective masses give rise to different effects compared to GaAs-based quantum dots: intrinsic exciton-exciton coupling, nonmolecular character of coupled dot exciton wave function, strong dependence of the oscillator strength on the dot height, large ground-state energy shift for dots separated by different barriers. Some of these effects make GaN/AlN quantum dots interesting candidates in quantum information processing.