What if a code does not automatically pop up on my Apple TV app?
How do I mirror to my Apple TV app when the code isn’t automatically popping up on my TV?
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Are you trying to AirPlay?
From what device?
To what device? You can AirPlay to an Apple TV box, or to select AirPlay-enabled TVs, or select AirPlay-enabled streaming devices. Not to other other devices. List of AirPlay 2–enabled smart TVs and third party streaming devices - Apple
AirPlay does not depend on the presence of the Apple TV app. Those are two separate functionalities.
Use AirPlay to stream video or mirror the screen of your iPhone, iPad, or iPod touch - Apple Support
If screen mirroring or streaming isn't working on your AirPlay-compatible device - Apple Support
AirPlay-enabled devices should be added to the Home app, and be assigned a room. Add a HomeKit accessory to the Home app - Apple Support