It does support background play. When you exit safari, you’ll notice it stops playing. Then, swipe down for control center, and click play on the music box. It should show the name of the video that was playing, like this.
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Click play, and the audio of your video will play in the background. Also, if you want PiP (Picture in Picture), when you are in full screen on your youtube video on the Web, open the app switcher and switch to another app WITHOUT going to the home screen. Then it will play in PiP in the other app you switched to.