This is the sequence I used to fix the exact same issue:
ONE: Make sure your Amazon Fire TV Cube (Gen.2) is updated. First, I Reset to Factory Default ( basically I started from scratch which might not need to be a step), Settings, My Fire TV, About, Check for Updates. Make sure you continue to download and install until it says "Your Fire TV is up to date" under Last Checked. I had to leave mine overnight before it said "Your Fire TV is up to date" (Jan.3, 2021, Current Version is Fire OS 7.2.2.9 (PS7229/1850)).
TWO: Restart your Amazon Fire TV Cube: Settings, My Fire TV, Restart
THREE: Clear Cashe, Clear data, then uninstall the app: (I don't know if you need to do it in sequence or not). Settings, Applications, Manage Installed Applications, Apple TV, Clear Cashe, Clear Data, Clear Data, Uninstall.
FOUR: Restart your Amazon Fire TV Cube again. Settings, My Fire TV, Restart
FIVE: Re-install the Apple TV App: Search, Apple TV, Install, Launch App
SIX: After entering the app I went into my account (Order at the top is Watch now, Library, Search, Account). I chose to "Sign-in using Apple ID" NOT "Sign-in using Mobile Device" not that that has anything to do with it, but it then asked for a Verification number from another Apple Device.
This was a two-day process for me and I had tried all of the steps I mentioned in different orders, but this was the ONLY order that worked for me. The fact that it took me two days may have been a factor so that the Amazon Fire TV Cube could download other "Stuff".
I hope this works for anyone else experiencing difficulties.
-Cheers!