We present an analysis of 20 tidal disruption events (TDEs) host galaxies observed with the Multi-Unit Spectroscopic Explorer (MUSE) integral-field spectrograph on European Southern Observatory (ESO) Very Large Telescope (VLT). We investigate the presence of extended emission line regions (EELRs) and study stellar populations mostly at sub-kpc scale around the host nuclei. EELRs are detected in 5/20 hosts, including two unreported systems. All EELRs are found at $z<0.045$, suggesting a distance bias and faint EELRs may be missed at higher redshift. EELRs only appear in post-merger systems and all such hosts at $z<0.045$ show them. Thus, we conclude that TDEs and galaxy mergers have a strong relation, and >45 per cent of post-merger hosts in the sample exhibit EELRs. Furthermore, we constrained the distributions of stellar masses near the central black holes (BHs), using the spectral synthesis code STARLIGHT and BPASS stellar evolution models. The youngest nuclear populations have typical ages of $\sim$1 Gyr and stellar masses below $2.5\,\mathrm{ M}_\odot$. The populations that can produce observable TDEs around non-rotating BHs are dominated by sub-solar-mass stars. 3/4 TDEs requiring larger stellar masses exhibit multipeaked light curves, possibly implying relation to repeated partial disruptions of high-mass stars. The found distributions are in tension with the masses of the stars derived using light curve models. Mass segregation of the disrupted stars can enhance the rate of TDEs from supersolar-mass stars but our study implies that low-mass TDEs should still be abundant and even dominate the distribution, unless there is a mechanism that prohibits low-mass TDEs or their detection.