My Apple TV app on the tv is treating it like I don’t have an existing account. I don’t want to sign up twice

The Apple Support pages do not list a limit per account for devices with the Apple TV app, but accounts are limited by number of simultaneous Apple TV+ streams.

You have to sign in to the Apple TV app with an Apple ID account that has the Apple TV+ subscription, or other Apple TV app Channel subscription, or iTunes Store purchases or rentals.

If the sign-in fails, then you may want to investigate your network.

Associated devices seem to be devices that allow downloading Apple DRM content, like Macs, iPhones, iPads. Managing my own associated devices does not list any streaming-only devices, like the Apple TV box.

It may not be relevant anymore, but pre 4th generation, we used to see a lot of people posting problems of Apple TV's being unable to use iTunes when folk tried to use them in quantity in motels etc.

At the time, I wondered if there may be some kind of temporary association while it was being used, I never had enough Apple TV's to test this

I find Apple documentation on associated devices, a little unclear.

Hehe 😉. Yes, I recall that you once wrote an interesting article on that.

Apple documentation states that ...

When you sign in with your Apple ID on a device and use that device to make purchases or subscribe to Apple services, the device becomes associated with your account.

You can have up to 10 associated devices (only 5 of which can be computers)

But, I don't believe, your issue is related to associations or other kinds of restriction by numbers.