You may have to wait a while if you want to use iOS 14’s picture-in-picture support to watch YouTube videos. MacRumors (via 9to5Mac) has noticed that YouTube’s mobile website stopped supporting iOS 14’s PIP mode, at least for free users. Try it and a video will only briefly pop out before it returns to the web page.
YouTube Premium subscribers have had success in using PIP, but it’s not clear if YouTube is deliberately restricting the feature. iPad users, who typically visit the desktop version of the website, haven’t had issues. The native YouTube app doesn’t currently support PIP on iPhone regardless of your Premium status.