Apple TV already logging me into newly downloaded app

I've been having troubles with one app

Please be specific about the app.

However, when I open the app I am already logged in

Apps that work with Single Sign-on will automatically sign you in based on your TV provider. Apps that work with Zero Sign-on will automatically sign you in based on your IP address from your ISP-TV provider.

When I try to reinstall the app I get the cloud and the arrow symbol instead of the "get" option, which also makes me think that I am not starting afresh after trying to delete and download the app.

When you try to reinstall the app and you get the cloud with arrow symbol instead of the "get" option, that mean that you have previously downloaded this app before, and you don’t need to “purchase” this item again. The difference is made by the App Store itself and your purchase history.

It says nothing about any remnants from the app from before deletion. Deleting an app on an iOS/iPadOS/tvOS device should take all its settings with it.