Absorption-selected galaxies offer an effective way to study low-mass galaxies at high redshift. However, the physical properties of the underlying galaxy population remain uncertain. In particular, the multiphase circumgalactic medium is thought to hold key information on gas flows into and out of galaxies that are vital for galaxy evolution models. Here, we present ALMA observations of CO molecular gas in host galaxies of H₂-bearing absorbers. In our sample of six absorbers, we detect molecular gas-rich galaxies in five absorber fields, although we did not target high-metallicity (>50 per cent solar) systems for which previous studies reported the highest detection rate. Surprisingly, we find that the majority of the absorbers are associated with multiple galaxies rather than single haloes. Together with the large impact parameters, these results suggest that the H₂-bearing gas seen in absorption is not part of an extended disc, but resides in dense gas pockets in the circumgalactic and intragroup medium....