Apparently the game still counts Barbas as a follower (I didn't killed him and took the mask) so this makes it impossible for me to have a new pet follower. Is there any way around this?

It's a known bug, and I don't think there's a fix for it on consoles.
You could try clearing the cache on the Xbox (this doesn't erase save files) and see if it helps.

Make sure the game is fully updated to the final patch, too. If you're connected online, it will prompt this automatically.