Why does Airplay stop playing and restarts every 45 minutes?

To steam using AirPlay from your iPhone, you need an AirPlay-compatible receiving device to stream to. That can be an Apple TV box, or an AirPlay-enabled smart TV, or an AirPlay-enabled third party streaming device. Without one of those, you wouldn’t be using AirPlay. Please specify.

Select smart TVs and select third party streaming devices support the Apple TV app. Then you wouldn’t need to use AirPlay for streaming content that is available in the Apple TV app. Then you may not experience the disconnect.

That seems to be a Fire TV Edition smart TV from the F30 series (model 2020/2021), which supports the Apple TV app. You can get the Apple TV app on there from the Amazon Fire TV Apps & Games Store. Get the Apple TV app or Apple TV+ app on your smart TV, streaming device, or game console - Apple Support

Download apps on your Fire TV device - Amazon Help & Customer Service

I mirror to Airplay

FireTVStick(Gen2),

AirPlay to what? AFAIK, Fire TV doesn’t support AirPlay.

AirPlay doesn’t have time limits, so the stream gets interrupted for other reasons.

I have a SmartTV

Can you find a model number on it?