The first SRG/eROSITA All-Sky Survey (eRASS1) provides the largest intracluster medium-selected galaxy cluster and group catalog covering the western Galactic hemisphere. Compared to samples selected purely on X-ray extent, the sample purity can be enhanced by identifying cluster candidates using optical and near-infrared data from the DESI Legacy Imaging Surveys. Using the red-sequence-based cluster finder eROMaPPer, we measured individual photometric properties (redshift z(lambda), richness lambda, optical center, and BCG position) for 12000 eRASS1 clusters over a sky area of 13 116 deg(2), augmented by 247 cases identified by matching the candidates with known clusters from the literature. The median redshift of the identified eRASS1 sample is z = 0.31, with 10% of the clusters at z > 0.72. The photometric redshifts have an accuracy of delta z/(1 + z) less than or similar to 0.005 for 0.05 < z < 0.9. Spectroscopic cluster properties (redshift z(spec) and velocity dispersion sigma) were measured a posteriori for a subsample of 3210 and 1499 eRASS1 clusters, respectively, using an extensive compilation of spectroscopic redshifts of galaxies from the literature. We infer that the primary eRASS1 sample has a purity of 86% and optical completeness >95% for z > 0.05. For these and further quality assessments of the eRASS1 identified catalog, we applied our identification method to a collection of galaxy cluster catalogs in the literature, as well as blindly on the full Legacy Surveys covering 24069 deg(2). Using a combination of these cluster samples, we investigated the velocity dispersion-richness relation, finding that it scales with richness as log(lambda(norm)) = 2.401 x log(sigma) - 5.074 with an intrinsic scatter of delta(in) = 0.10 +/- 0.01 dex. The primary product of our work is the identified eRASS1 cluster catalog with high purity and a well-defined X-ray selection process, opening the path for precise cosmological analyses presented in companion papers.