Context. This paper presents new results from the ESO Large Programme Looking into the faintEst WIth MUSE (LEWIS). The LEWIS sample consists of low-surface-brightness galaxies (LSBs) and ultra-diffuse galaxies (UDGs) located inside 0.4Rvir of the Hydra I cluster. Integral-field spectroscopy is acquired for 24 galaxies with the MUSE spectrograph mounted on the Very Large Telescope (VLT). Aims. Our main objectives are to analyse possible correlations between the environment and the integrated stellar population properties of our targets, based on which we infer clues about their formation. Methods. For each galaxy in the sample, we extracted the 1D stacked spectrum in an aperture of one effective radius Re and adopted previously published stellar kinematics to derive the age, metallicity, and [Mg/Fe] through a full spectral fitting technique. Results. We find that the analysed LEWIS sample has a mean metallicity of ⟨[M/H]⟩= − 0.9 ± 0.2 dex and a mean age of 10 ± 2 Gyr, comparable to previous results of UDGs in other clusters. According to their position in the projected phase space, galaxies can be classified into two groups: very early infaller galaxies, which on average have slightly higher metallicities (⟨[M/H]⟩early = −0.8 ± 0.1 dex), and late infaller galaxies, with slightly lower values (⟨[M/H]⟩late = −1.0 ± 0.1 dex). According to their properties, late-infallers tend to be rotation-supported systems. Conversely, two types of galaxies are found in the early-infall region. Roughly half have metallicities consistent with the dwarf galaxy mass─metallicity relation. The other half show higher metallicities (with ⟨[M/H]⟩≥ − 1.0 dex) and are located outside the 1σ scatter of the mass-metallicity relation. The two subgroups of early-infallers also display different timescales for stellar mass assembly. Metal-rich galaxies reached 50% of their stellar mass in less than 1 Gyr and show a prolonged and almost constant star formation over more than 12 Gyr. The other galaxies exhibit a star formation history similar to that found for galaxies in the late-infall region. Both early and late infallers show solar-like α abundances. Conclusions. From the analysis of stellar population properties presented in this work and of stellar kinematics previously obtained from LEWIS, we identified different classes of UDGs within the Hydra I cluster ─ as shown by metallicities, quenching timescales, and kinematics ─ which suggest different formation mechanisms. Almost all of the UDGs and LSBs in this cluster are consistent with the puffed-up dwarf formation scenario, having dwarf-like metallicities and being consistent with the mass-metallicity relation for dwarfs. In the innermost regions of the cluster, where more metal-rich UDGs are found, tidal effects or the environment might have influenced their formation and evolution.