Hello @TheOtherKai, first of - thanks for taking the time for properly reporting this. Steps provided is greatly appreciated!
I've looked into it and it's definitely a thing. I went and investigated WebKit APIs as well as sources and didn't found a way to instruct it which display to put PiP window onto. Moreower - at this time it behaves exactly the same as in Safari (record uses DeskPad because I wasn't able to record both screens at the same time otherwise):
Safari might improve or fix that WebKit behaviour in future so I'm marking this as NoFix for now. In case we figure something out - I'll definitelly get back here.