Hello uLu_MuLu,
We have a great support article that might help with this. Please take a look at it here: Play web videos in Safari on Mac - Apple Support. Make sure these are the steps you are following to use the Picture in Picture feature.
If you are following that guidance, we advise that you restart your Mac (Shut down or restart your Mac) and make sure it's fully updated (Update macOS on Mac). Once those are done, please test this issue.
If the issues persist, continue by testing to see when this behavior occurs. First, put your Mac into safe mode and test. Booting into safe mode will disable any startup and login items from automatically opening, which can help us to isolate the source of the issue further. When you do this, you’ll see some unusual behavior from the display; this is to be expected. Please follow the steps in this article: How to use safe mode on your Mac.
Next, let's see if this behavior persists in a new user account on your Mac. Testing in a new user will determine if you have a system-wide issue or if it's user-specific. Please follow these steps to make a test account that can be deleted later: Change Users & Groups settings on Mac. Once the new user account has been set up, log out of your own account then into the new one to test the issue.
Let us know how this goes, please. Thanks and take care.