I think I also had this issue using Cinnamon once, but then I just used VLC instead. Never bothered to look into why. Worked fine in GNOME for me though.
I found a thread with a similar issue: https://forums.opensuse.org/t/flatpak-mpv-broken-since-20240306-snapshot/172981
There it seems the issue was audio. Try running the flatpak version with
flatpak run io.mpv.Mpv --ao=pulse path-to-your-media-file
If the issue is with the DE, I’m actually using the Xfce edition of Linux Mint. I would just use VLC but it gives me performance issues because I don’t have the best hardware and mpv seems to work much more efficiently. But yeah, changing the audio output fixed this issue, I’m guessing the flatpak version of mpv defaults to pipewire. I was curious and I did test pipewire with both versions and I got the same results that they did.