  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 ($\approx$ 60% of
the total Swift short GRB population). We contribute 245 new multi-band imaging
observations across 49 distinct GRBs and 25 spectra of their host galaxies.
Supplemented by literature and archival survey data, the catalog contains 335
photometric and 40 spectroscopic data sets. The photometric catalog reaches
$3\sigma$ depths of $\gtrsim 24-27$ mag and $\gtrsim 23-26$ mag for the optical
and near-infrared bands, respectively. We identify host galaxies for 84 bursts,
in which the most robust associations make up 54% (49/90) of events, while only
a small fraction, 6.7%, have inconclusive host associations. Based on new
spectroscopy, we determine 17 host spectroscopic redshifts with a range of
$z\approx 0.15-1.6$ and find that $\approx$ 25-44% of Swift short GRBs
originate from $z>1$. We also present the galactocentric offset catalog for 83
short GRBs. Taking into account the large range of individual measurement
uncertainties, we find a median of projected offset of $\approx 7.9$ kpc, for
which the bursts with the most robust associations have a smaller median of
$\approx 4.9$ kpc. 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 BRIGHT website.
