  Optical excitations in moir\'e transition metal dichalcogenide bilayers lead
to the creation of excitons, as electron-hole bound states, that are
generically considered within a Bose-Hubbard framework. Here, we demonstrate
that these composite particles obey an angular momentum commutation relation
that is generally non-bosonic. This emergent spin description of excitons
indicates a limitation to their occupancy on each site, which is substantial in
the weak electron-hole binding regime. The effective exciton theory is
accordingly a spin Hamiltonian, which further becomes a Hubbard model of
emergent bosons subject to an occupancy constraint after a Holstein-Primakoff
transformation. We apply our theory to three commonly studied bilayers
(MoSe2/WSe2, WSe2/WS2, and WSe2/MoS2) and show that in the relevant parameter
regimes their allowed occupancies never exceed three excitons. Our systematic
theory provides guidelines for future research on the many-body physics of
moir\'e excitons.
