  We revisit and extend the standard bosonic interpretation of interlayer
excitons in the moir\'e potential of twisted heterostructures of
transition-metal dichalcogenides. In our experiments, we probe a high quality
MoSe$_2$/WSe$_2$ van der Waals bilayer heterostructure via density-dependent
photoluminescence spectroscopy and reveal strongly developed, unconventional
spectral shifts of the emergent moir\'e exciton resonances. The observation of
saturating blueshifts of successive exciton resonances allow us to explain
their physics in terms of a model utilizing fermionic saturable absorbers. This
approach is strongly inspired by established quantum-dot models, which
underlines the close analogy of interlayer excitons trapped in pockets of the
moir\'e potential, and quantum emitters with discrete eigenstates.
