Gravitational waves (GWs) from compact binary coalescences (CBCs) can constrain the cosmic expansion of the universe. In the absence of an associated electromagnetic counterpart, the spectral sirens method exploits the relation between the detector frame and the source frame masses to jointly infer the parameters of the mass distribution of black holes (BH) and the cosmic expansion parameter $H_0$. This technique relies on the choice of the parametrization for the source mass population of BHs observed in binary black holes merger (BBHs). Using astrophysically motivated BBH populations, we study the possible systematic effects affecting the inferred value for $H_0$ when using heuristic mass models like a broken power law, a power law plus peak and a multi-peak distributions. We find that with 2000 detected GW mergers, the resulting $H_0$ obtained with a spectral sirens analysis can be biased up to $3\sigma$. The main sources of this bias come from the failure of the heuristic mass models used so far to account for a possible redshift evolution of the mass distribution and from their inability to model unexpected mass features. We conclude that future dark siren GW cosmology analyses should make use of source mass models able to account for redshift evolution and capable to adjust to unforeseen mass features.