My Apple TV stopped recognizing my iPhone or vice versa.

If your smart TV is AirPlay-enabled, then you can AirPlay content to it. Other models will not have this functionality.

You may need to add this TV to your Home in the Home.app. There you can configure access.

The Apple TV app on your smart TV is a different feature, and quite independent of AirPlay.

List of AirPlay 2-enabled smart TVs - Apple

If you were referring to other types of recognition, then please elaborate.

The Netflix update of January 20th, seems to have requirements for the Apple TV system software to be updated as well. How to update your Apple TV - Apple Support

I am not aware of issues from December.