We present a comprehensive optical and near-infrared census of the fields of 90 short gamma-ray bursts (GRBs) discovered in 2005-2021, constituting all short GRBs for which host galaxy associations are feasible (~60% of the total Swift short GRB population). We contribute 274 new multi-band imaging observations across 58 distinct GRBs and 26 spectra of their host galaxies. Supplemented by literature and archival survey data, the catalog contains 542 photometric and 42 spectroscopic data sets. The photometric catalog reaches 3{sigma} depths of >~24-27mag and >~23-26mag for the optical and near-infrared bands, respectively. We identify host galaxies for 84 bursts, in which the most robust associations make up 56% (50/90) of events, while only a small fraction, 6.7%, have inconclusive host associations. Based on new spectroscopy, we determine 18 host spectroscopic redshifts with a range of z~0.15-1.5 and find that ~23%-41% of Swift short GRBs originate from z>1. We also present the galactocentric offset catalog for 84 short GRBs. Taking into account the large range of individual measurement uncertainties, we find a median of projected offset of ~7.7kpc, for which the bursts with the most robust associations have a smaller median of ~4.8kpc. Our catalog captures more high-redshift and low-luminosity hosts, and more highly offset bursts than previously found, thereby diversifying the population of known short GRB hosts and properties. In terms of locations and host luminosities, the populations of short GRBs with and without detectable extended emission are statistically indistinguishable. This suggests that they arise from the same progenitors, or from multiple progenitors, which form and evolve in similar environments. All of the data products are available on the Broadband Repository for Investigating Gamma-Ray Burst Host Traits (BRIGHT) website.