Hello or_09,
Thanks for that info and for choosing the Apple Support Communities. If we understand correctly, you are unable to use your AirPods with your Sony TV. If you have already checked resources from Sony that this TV can output audio to a Bluetooth device and this works with other Bluetooth audio devices, make sure you are following the steps from this Apple resource Set up AirPods with your Mac and other Bluetooth devices:
"Use AirPods with a non-Apple device
You can use AirPods as a Bluetooth headset with a non-Apple device. You can't use Siri, but you can listen and talk. To set up your AirPods with an Android phone or other non-Apple device,3 follow these steps:
1. On your non-Apple device, go to the settings for Bluetooth and make sure that Bluetooth is on.4 If you have an Android device, go to Settings > Connections > Bluetooth.
2. With your AirPods in the charging case, open the lid.
3. Press and hold the setup button on the back of the case until the status light flashes white.
4. When your AirPods appear in the list of Bluetooth devices, select them."
If you are still having an issue, please reset your AirPods and set them up again: How to reset your AirPods and AirPods Pro
Cheers!