Studying galaxies at different cosmic epochs entails several observational effects that need to be taken into account to compare populations across a large time-span in a consistent manner. We use a sample of 166 nearby galaxies that hosted type Ia supernovae (SNe Ia) and have been observed with the integral field spectrograph MUSE as part of the AMUSING survey. Here, we present a study of the systematic errors and bias on the host stellar mass with increasing redshift, which are generally overlooked in SNe Ia cosmological analyses. We simulate observations at different redshifts (0.1 < z < 2.0) using four photometric bands (griz, similar to the Dark Energy Survey-SN program) to then estimate the host galaxy properties across cosmic time. We find that stellar masses are systematically underestimated as we move towards higher redshifts, due mostly to different rest-frame wavelength coverage, with differences reaching 0.3 dex at z similar to 1. We used the newly derived corrections as a function of redshift to correct the stellar masses of a known sample of SN Ia hosts and derive cosmological parameters. We show that these corrections have a small impact on the derived cosmological parameters. The most affected is the value of the mass step Delta(M), which is reduced by similar to 0.004 (6% lower). The dark energy equation of state parameter w changes by Delta w similar to 0.006 (0.6% higher) and the value of Omega(m) increases at most by 0.001 (similar to 0.3%), all within the derived uncertainties of the model. While the systematic error found in the estimate of the host stellar mass does not significantly affect the derived cosmological parameters, it is an important source of systematic error that needs to be corrected for as we enter a new era of precision cosmology.