If you use KDE, you can work around this because of the powerful feature set the window manager has for setting custom window behavior.

1. Right click the PIP window and then click "More Actions-> Special Window Settings".

2. On the window that pops up, click "Add Property", and add "Window title". Change the drop-down from "Unimportant" to "Exact match" (this works on Firefox because the window title is always "Picture-in-Picture", you might have to do something slightly different on Chrome if it does something different).

3. Click "Add Property" again, add "Keep above other windows", change the drop-down to "Force", and change the radio button to "Yes".

4. From now on, all PIP windows will show up on top of other windows.

It would definitely be nicer if there was some sort of "always on top" permission that applications could request, but it's not too bad.

For Chrome it's "Picture in picture"