signing into Apple TV app

There may be an issue with your account (e.g. incomplete user information, or invalid or expired payment method).

More precise messages may be available in the iTunes application (Windows, older Mac), or on iOS/​iPadOS Settings, or macOS TV app/​Music app.

I no longer have an iTunes account per se

It’s officially your Apple ID account, not just for iTunes. And those accounts are quite permanent.

and have deleted the app.

So re-download it.

I wonder if it is related to the fact that my mac did not update itself for a year or two.

Maybe. The Mac App Store (which is used for OS updates too) also requires that you sign in with your Apple ID account. Depending on the Mac model and age, it may have already had its highest supported OS version?

So maybe the apple TV thinks I have iTunes?

The message is about using the iTunes application (or similar) to manage your account and correct/​update/​complement the needed details. The message makes no assumption of what apps or accounts you already have.