Apple TV 2nd gen cannot login with 2-factor authorization

You can't turn off two-factor authentication. This is a problem for this generation of Apple TV units because they were built between 2010 and 2012, while Apple adopted 2FA about 2015. Therefore the code has to be retro fitted to function by entering it as part of your password.

When you have the code enter the six digits immediately after your password and it should work.

I have to add though that as this hardware is obsolete the time has come to replace it with a new one.

I saw a very useful tip here by Urquhart1244, a senior member of this forum, which advised to type the password then request a new verification code. This takes the pressure off getting all the characters in place before the code expires.

Please note that on legacy devices, you should be able to sign in for iTunes Store services, but not for iCloud services. The latter is unfortunately no longer available on these models. Also see: discussions.apple.com/thread/255664587