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We revisit and extend the standard bosonic interpretation of interlayer excitons in the moiré potential of twisted heterostructures of transition-metal dichalcogenides. In our experiments, we probe a high quality MoSe2/WSe2 van der Waals bilayer heterostructure via density-dependent photoluminescence spectroscopy and reveal strongly developed, unconventional spectral shifts of the emergent moiré 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 moiré potential, and quantum emitters with discrete eigenstates.