The good news is that if this is something you were actually looking forward to, YouTube has since announced that they will soon be making the feature available to all users, both Premium and non-Premium subscribers. However, oddly enough, the company says that this will only be limited to those living in the US.
In a statement made to MacRumors, a YouTube spokesperson was quoted as saying, “Picture-in-Picture (PiP) allows users to watch YouTube videos in a small mini player while simultaneously browsing outside of the YouTube app on their mobile device. We’re starting to roll out PiP for YouTube Premium members on iOS and plan to launch PiP for all US iOS users as well.”
It seems odd that this would be geo-restricted since it doesn’t seem to have anything to do with licensing content which could sometimes be geo-restricted, but hopefully YouTube will expand the feature to non-US users in the future.