ABSTRACT We present an analysis of young star clusters (YSCs) that form in the E-MOSAICS cosmological, hydrodynamical simulations of galaxies and their star cluster populations. Through comparisons with observed YSC populations, this work aims to test models for YSC formation and obtain an insight into the formation processes at work in part of the local galaxy population. We find that the models used in E-MOSAICS for the cluster formation efficiency and high-mass truncation of the initial cluster mass function ($M_\rm {c,* }$) both quantitatively reproduce the observed values of cluster populations in nearby galaxies. At higher redshifts (z ≥ 2, near the peak of globular cluster formation) we find that, at a constant star formation rate (SFR) surface density, $M_\rm {c,* }$ is larger than at z = 0 by a factor of four due to the higher gas fractions in the simulated high-redshift galaxies. Similar processes should be at work in local galaxies, offering a new way to test the models. We find that cluster age distributions may be sensitive to variations in the cluster formation rate (but not SFR) with time, which may significantly affect their use in tests of cluster mass-loss. By comparing simulations with different implementations of cluster formation physics, we find that (even partially) environmentally independent cluster formation is inconsistent with the brightest cluster-SFR and specific luminosity-$Σ _\rm {SFR}$ relations, whereas these observables are reproduced by the fiducial, environmentally varying model. This shows that models in which a constant fraction of stars form in clusters are inconsistent with observations.