We examine the properties of damped Lyman-$alpha$ absorbers (DLAs) emerging from a single set of cosmological initial conditions in two state-of-the-art cosmological hydrodynamic simulations: {sc Simba} and {sc Technicolor Dawn}. The former includes star formation and black hole feedback treatments that yield a good match with low-redshift galaxy properties, while the latter uses multi-frequency radiative transfer to model an inhomogeneous ultraviolet background (UVB) self-consistently and is calibrated to match the Thomson scattering optical depth, UVB amplitude, and Ly-$alpha$ forest mean transmission at $z>5$. Both simulations are in reasonable agreement with the measured stellar mass and star formation rate functions at $zgeq 3$, and both reproduce the observed neutral hydrogen cosmological mass density, $Omega_{rm HI}(z)$. However, the DLA abundance and metallicity distribution are sensitive to the galactic outflows feedback and the UVB amplitude. Adopting a strong UVB and/or slow outflows under-produces the observed DLA abundance, but yields broad agreement with the observed DLA metallicity distribution. By contrast, faster outflows eject metals to larger distances, yielding more metal-rich DLAs whose observational selection may be more sensitive to dust bias. The DLA metallicity distribution in models adopting an ${rm H}_2$-regulated star formation recipe includes a tail extending to $[M/H] ll -3$, lower than any DLA observed to date, owing to curtailed star formation in low-metallicity galaxies. Our results show that DLA observations play an imporant role in constraining key physical ingredients in galaxy formation models, complementing traditional ensemble statistics such as the stellar mass and star formation rate functions.
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