Are captions available on Apple TV?

That option is available in the YouTube app on Apple TV as well:

In the YouTube app on Apple TV, during playback, press Up or Pause to bring up the extra in-video controls. Navigate to the gears symbol for Settings there. In those settings, go to Captions. In the Captions settings, select (✓) “English (auto-generated)”, if so wanted.

On Apple Store there is only one app for all the Apple devices.

It may look like that, it is administrated like that, but there is a separately compiled edition for each platform (iOS/​iPadOS/​tvOS). The app editions are not really the same. Their version numbering doesn’t even have to match.

Looks like it is a programming bug.

Maybe. How to contact an app developer - Apple Support

AFAIK, it has to be added to the source stream by the original video uploader. It will not be available to just any video.

The setting on desktop browser is “Include auto-generated captions (when available)”. Notice the “when available”.

Ask Google.

It is their service and their app.