There is no Apple TV app available on my TV

Apple’s protocol for local streaming (AirPlay) is different from Chromecast and other casting protocols. Your iPhone would support AirPlay on a device level, but other casting protocols depend on what each app may have added.

Apple is trying to get TV manufacturers to include AirPlay capabilities, including Sony.

Does my TV support Apple AirPlay 2? - Sony Support

Streaming from the Apple TV app requires an AirPlay-streaming-from device (e.g. iPhone) and an AirPlay-streaming-to device (e.g. AirPlay-enabled TV or Apple TV set-top box). If your TV doesn’t have AirPlay capabilities, then you could consider the Apple TV set-top box to add AirPlay capability through HDMI.

Chromecast is Google's own proprietary streaming protocol, and would require Google allow Apple to use it for their own Apps and media, which is likely not the case.

There are some Sony Tvs that support Airplay, and you should be able to use that to Airplay Aplpe Tv+ content to it.

Apple TV app and Airplay 2 compatibility - Devices - Apple

• Sony Z9G Series (2019)
• Sony A9G Series (2019)
• Sony X950G Series (2019)
• Sony X850G Series (2019 85", 75", 65" and 55" models)

Content that can be Airplayed is controlled by the content provider. Not all providers all Airplay to work with their content.

Please follow the directions below to use Airplay:

https://support.apple.com/en-us/HT204289