I dont know about iOS browsers i only used Safari, but seems like Brave does, so i guess it should be technically possible
Every desktop browser works in this way tho.
Imagine you are watching a casual youtube video with the news or whatever, and you quickly want to send a simple text message. you would have to fullscreen the video, then press picture in picture to have it not stop while you do this simple task.
and Picture in Picture is not ideal on smartphone anyway since you have way less screen space.
Even while writting this on a desktop i have a video running in the background, i would get pretty anyoed if it would stop each time i had to do something quick outside the browser app