Can I pair AirPods with Apple TV Model 1625 released in 2015?
Yes, that will work with Apple TV (4th generation or later).
The first time you do it: set the AirPods to pairing mode (AirPods in case with open lid, near device to pair with), go to the bluetooth screen on Apple TV, select the discovered headphones, and pair and connect. Any time after that, the headphones will be remembered and can be connected to without pairing again, and selected as audio output, e.g. in Apple TV control Center.
If the AirPods are registered with the same Apple ID account as the Apple TV, then Apple TV may alert you about “nearby AirPods” that are connectable, as a shortcut, to make it slightly easier to connect.
When you use AirPlay to stream from iPhone to Apple TV, then the defaults of the Apple TV box apply. iPhone headphones will then not be the audio output anymore. Set headphones on Apple TV beforehand or during playback to Apple TV.