Recent systematic searches for massive black holes (BHs) in local dwarf galaxies led to the discovery of a population of faint active galactic nuclei (AGNs). We investigate the agreement of the BH and AGN populations in the Illustris, TNG, Horizon-AGN, EAGLE, and SIMBA simulations with current observational constraints in low-mass galaxies. We find that some of these simulations produce BHs that are too massive, and that the BH occupation fraction (OF) at z = 0 is not inherited from the simulation seeding modelling. The ability of BHs and their host galaxies to power an AGN depends on BH and galaxy subgrid modelling. The fraction of AGN in low-mass galaxies is not used to calibrate the simulations, and thus can be used to differentiate galaxy formation models. AGN fractions at z = 0 span two orders of magnitude at fixed galaxy stellar mass in simulations, similarly to observational constraints, but uncertainties and degeneracies affect both observations and simulations. The agreement is difficult to interpret due to differences in the masses of simulated and observed BHs, BH OF affected by numerical choices, and an unknown fraction of obscured AGN. Our work advocates for more thorough comparisons with observations to improve the modelling of cosmological simulations, and our understanding of BH and galaxy physics in the low-mass regime. The mass of BHs, their ability to efficiently accrete gas, and the AGN fraction in low-mass galaxies have important implications for the build-up of the entire BH and galaxy populations with time.