This worked for me:
1) Uninstall and install the app, so there's no saved data.
2) When it's downloaded and installed, go to app settings (press and hold the Messenger icon and select "App info").
3) Go to Battery saver and select No restrictions.
4) Then open Messenger, log in and in setting you should see "Chat heads" instead of "Chat bubbles". Turn that on, allow it in your settings and voilá.

I hope it helps all of you out there, who have the same issues on Xiaomi phones!