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We present a comparison of galaxy atomic and molecular gas properties in three recent cosmological hydrodynamic simulations, Simba, EAGLE, and Illustris-TNG, versus observations from $z\sim 0-2$. These simulations all rely on similar sub-resolution prescriptions to model cold interstellar gas which they cannot represent directly, and qualitatively reproduce the observed $z\approx 0$ HI and H$_2$ mass functions (HIMF, H2MF), CO(1-0) luminosity functions (COLF), and gas scaling relations versus stellar mass, specific star formation rate, and stellar surface density $μ_*$, with some quantitative differences. To compare to the COLF, we apply an H$_2$-to-CO conversion factor to the simulated galaxies based on their average molecular surface density and metallicity, yielding substantial variations in $α_{\rm CO}$ and significant differences between models. Using this, predicted $z=0$ COLFs agree better with data than predicted H2MFs. Out to $z\sim 2$, EAGLE's and Simba's HIMF and COLF strongly increase, while TNG's HIMF declines and COLF evolves slowly. EAGLE and Simba reproduce high $L_{\rm CO1-0}$ galaxies at $z\sim 1-2$ as observed, owing partly to a median $α_{\rm CO}(z=2)\sim 1$ versus $α_{\rm CO}(z=0)\sim 3$. Examining \HI, H$_2$, and CO scaling relations, their trends with $M_*$ are broadly reproduced in all models, but EAGLE yields too little HI in green valley galaxies, TNG and Simba overproduce cold gas in massive galaxies, and Simba overproduces molecular gas in small systems. Using Simba variants that exclude individual AGN feedback modules, we find that Simba's AGN jet feedback is primarily responsible by lowering cold gas contents from $z\sim 1\to0$ by suppressing cold gas in $M_*> 10^{10}{\rm M}_\odot$ galaxies, while X-ray feedback suppresses the formation of high-$μ_*$ systems.