Hello Arthurw99,
Thank you for posting to Apple Support Communities.
We're happy to help. We understand you're trying to get the picture-in-picture to work in Safari on iPhone. Since this has happened, have you contacted YouTube support at all? They should have an answer as to whether this is possible with, or without, a subscription to the service.
From the iPhone end, make sure that iOS is updated. Before updating, we recommend backing up your iPhone first. These articles can be of help. When you back up, you can back up using a Mac or PC rather than iCloud if you choose.
How to back up your iPhone, iPad, and iPod touch
Update iOS on iPhone
If the issue continues, we recommend contacting YouTube support or Apple Support directly. You can contact Apple Support here: Contact Apple Support
Take care.