I did not find a setting to activate it.
This is not possible without a third-party application. YouTube did not add the ability to iOS. You can suggest it to YouTube, but Apple has nothing to do with it.
Are there any settings I can use or third-party apps that I can use?
There is an app called PiPifier. Link to download This allows the playback of videos on the Web in picture-in-picture.
How to use it
If you're on a YouTube video on the web, go to the sharing menu on the top right:
Then click on more:
If you're in a YouTube video on the web, click PiPifier: