FaceTime on Xbox: Just Works
A Reddit user denounces the situation. He is able to make FaceTime calls by connecting his Xbox to a Logitech C930 webcam. To join a FaceTime call, he opens the Microsoft Edge browser on the Edge box and clicks a FaceTime call invitation to join. Starting a FaceTime call yourself is not possible. It is possible on any device with Google Chrome or Microsoft Edge, thanks to the FaceTime links that allow you to invite someone for a conversation. This is primarily intended for Windows and Android users, but will work on many more devices.
You can't make FaceTime calls on Apple TV because there's no FaceTime app
or web browser is present. In addition, the ports to connect a camera are missing. With the latest version of the Apple TV, which came out last fall, Apple could have included it but didn't. What is possible is to show a FaceTime conversation on your television screen via AirPlay. But the video still comes from your iPhone or iPad.
A Reddit user's setup.
FaceTime with Microsoft products
There are rumors that Apple is working on a combination of Apple TV and HomePod that will allow you to make FaceTime calls thanks to a built-in camera. Apple added FaceTime and iMessage support to the software running the HomePod in its code. But there is no such device yet.
If you want to FaceTime on your television, you can do that thanks to the Xbox and Edge browsers, both from Microsoft.