Photoluminescence (PL) spectroscopy performed on arrays of semiconductor nanowires (NWs) suffers from ensemble broadening of PL lines, and fails to separate the PL from NWs of different crystal structures in the ensemble. Even the results on PL from single NWs are not devoid of ambiguity. This is because the influence of structural defects in NWs, such as stacking faults, twin boundaries and dislocations, on their optical spectra cannot be accounted for since the structural characteristics of the same NW remain largely unknown. We performed low-temperature PL spectroscopy on individual wurtzite (WZ) ZnSe NWs, and confirmed a homogeneous single-crystalline microstructure without any extended defects in these NWs, thus excluding any role of structural imperfections in their optical spectra. The luminescence is shown to be dominated solely by native point defects, while no role of extrinsic impurities was found. The radiative recombination is shown to originate from excitons bound to vacancies of Zn (VZn), VZn-complexes, and their phonon replicas. The binding energies of the acceptor-bound excitons, ionization energies of the acceptors, and average number of phonons emitted for shallow donor-VZn acceptor pair related transition were determined. Distinct from previous studies on PL from arrays of ZnSe NWs, this work provides an unambiguous interpretation of the PL spectra and assignment of PL peaks to WZ ZnSe. Narrow excitonic emission of linewidths 2.9 meV indicate excellent optical quality of WZ ZnSe NWs.