Apple TV and Icloud
Why can’t I sign into my iCloud account on my Apple TV? It works on all my other devices. My software is up to date.
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When signing in with 2 factor authentication on an older device you need to enter the password and code together, and there is a time limit. To avoid exceeding the limit enter the password, then request the code, and then add it to the end of the password
Get a verification code and sign in with two-factor authentication - Apple Support