/media is for removable drives. If you mount something there, file managers like Gnome will show you the “eject” or “disconnect” button.
/mnt drives show up as regular network drives without that “eject” functionality.
I don’t know if I’m doing something wrong, but I have a secondary SSD in my laptop that I mount on /mnt/elyssa
and in every DE and distro I tried it appeared as a removable drive with the “eject” button. Right now I use Fedora with Gnome and if I install this extension or enable the removable drives option in Dash to Dock, it shows me that drive. Maybe some mount option in Gnome Disks, but since it’s not that big of a problem, I haven’t looked too much into it.