I am currently on Fedora 35 Workstation and using Firefox PiP on YouTube(any media that allows picture-in-picture) does not have a separate window or player when alt-tabbing. I tested Microsoft Edge for Linux and picture-in-picture player has a dedicated window. I am not sure if this is a limitation on Linux, but I think this feature already exists on the Windows version where it is possible to alt-tab between Firefox PiP windows.