In this second paper of the series we study , with new Keck/DEIMOS spectra , the stellar populations of seven spectroscopically confirmed ultra-diffuse galaxies ( UDGs ) in the Coma cluster . We find intermediate to old ages ( \sim 7 Gyr ) , low metallicities ( [ Z/H ] \sim – 0.7 dex ) and mostly super-solar abundance patterns ( [ Mg/Fe ] \sim 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 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 .