AirPlay
How do I stream something to Apple TV then continue using my phone without it cutting off or playing another video that comes up on my phone
iPhone X, iOS 14
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For apps that support in-app AirPlay, you can switch between apps without interrupting the stream, unless that other app is also set to use AirPlay to the Apple TV by default. I.e. you can switch to WhatsApp, but maybe not to YouTube.
For half-supported apps that rely on Screen Mirroring for AirPlay, you must stay with that app, or it will just mirror the other thing that you switched to.
In general, if you want to do “whatever” with your iPhone, then you shouldn’t use your iPhone for AirPlay at that time, as it may be locked to that use.
Use AirPlay to stream video or mirror the screen of your iPhone, iPad, or iPod touch - Apple Support