We discuss the nature and physical properties of gas-mass selected galaxies in the ALMA spectroscopic survey (ASPECS) of the Hubble Ultra Deep Field (HUDF). We capitalize on the deep optical integral-field spectroscopy from the MUSE HUDF Survey and multi-wavelength data to uniquely associate all 16 line-emitters, detected in the ALMA data without preselection, with rotational transitions of carbon monoxide (CO). We identify ten as CO(2-1) at \(1 < z < 2\), five as CO(3-2) at \(2 < z < 3\) and one as CO(4-3) at \(z = 3.6\). Using the MUSE data as a prior, we identify two additional CO(2-1)-emitters, increasing the total sample size to 18. We infer metallicities consistent with (super-)solar for the CO-detected galaxies at \(z \le 1.5\), motivating our choice of a Galactic conversion factor between CO luminosity and molecular gas mass for these galaxies. Using deep Chandra imaging of the HUDF, we determine an X-ray AGN fraction of 20% and 60% among the CO-emitters at \(z \sim 1.4\) and \(z \sim 2.6\), respectively. Being a CO-flux limited survey, ASPECS-LP detects molecular gas in galaxies on, above and below the main sequence (MS) at \(z \sim 1.4\). For stellar masses \(\ge 10^{10} (10^{10.5})\) M\(_{\odot}\), we detect about 40% (50%) of all galaxies in the HUDF at \(1 < z < 2\) (\(2 < z < 3\)). The combination of ALMA and MUSE integral-field spectroscopy thus enables an unprecedented view on MS galaxies during the peak of galaxy formation.