Also, to see the ‘Apple TV Remote’ in the Control Center you may need to customize the Control Center in the system Settings app.
Here is more info: https://support.apple.com/guide/tv/use-ios-or-ipados-control-center-atvb701cadc1/tvOS
You don’t seem to need the App installed to use it this way.
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So I checked my Wife's phone and she does indeed have Apple TV Keyboard under notifications. Looks like it does rely on using the same AppleID then to get the notifications.