Several astrophysical scenarios have been proposed to explain the origin of the population of binary black hole (BBH) mergers detected in gravitational waves (GWs) by the LIGO/Virgo Collaboration. Among them, BBH mergers assembled dynamically in young massive and open clusters have been shown to produce merger rate densities consistent with LIGO/Virgo estimated rates. We use the results of a suite of direct, high-precision $N$-body evolutionary models of young massive and open clusters and build the population of BBH mergers, by accounting for both a cosmologically-motivated model for the formation of young massive and open clusters and the detection probability of LIGO/Virgo. We show that our models produce dynamically-paired BBH mergers that are well consistent with the observed masses, mass ratios, effective spin parameters, and final spins of the second Gravitational Wave Transient Catalog (GWTC-2).