In this second paper of the series we study, with new Keck/DEIMOS spectra, the stellar populations of seven spectroscopically confirmed ultradiffuse galaxies (UDGs) in the Coma cluster. We find intermediate to old ages (∼7 Gyr), low metallicities ([Z/H] ∼ −0.7 dex) and mostly supersolar abundance patterns ([Mg/Fe] ∼ 0.13 dex). These properties are similar to those of low-luminosity (dwarf) galaxies inhabiting the same area in the cluster and are mostly consistent with being the continuity of the stellar mass scaling relations of more massive galaxies. These UDGs’ star formation histories imply a relatively recent infall into the Coma cluster, consistent with the theoretical predictions for a dwarf-like origin. However, considering the scatter in the resulting properties and including other UDGs in Coma, together with the results from the velocity phase-space study of the Paper I in this series, a mixed-bag of origins is needed to explain the nature of all UDGs. Our results thus reinforce a scenario in which many UDGs are field dwarfs that become quenched through their later infall onto cluster environments, whereas some UDGs could be genuine primordial galaxies that failed to develop due to an early quenching phase. The unknown proportion of dwarf-like to primordial-like UDGs leaves the enigma of the nature of UDGs still open.