Apple Tv 3

Apple TV 3rd generation is not capable of two factor authentication. Instead, there is another procedure for it to pass this hurdle.

Apple explains this in Two-factor authentication for Apple ID

What if I use two-factor authentication on a device running older software?

If you use two-factor authentication with devices running older OS versions—like an Apple TV (2nd or 3rd generation)—you might be asked to add your six-digit verification code to the end of your password when signing in. Get your verification code from a trusted device running iOS 9 and later or OS X El Capitan and later, or have it sent to your trusted phone number. Then type your password followed by the six-digit verification code directly into the password field.

It seems that the official support answer, reproduced above, may not work for all users, because of unmentioned time limits. Instead, I’d like to offers this:

If you have an Apple TV (2nd or 3rd generation) and two-factor authentication for your Apple ID, then you may be asked to type your password followed by the six-digit verification code directly into the password field. Additionally, there is a time limit, after which the code becomes invalid.

Try this: First enter the characters of your password (do not submit yet), then manually request a new verification code, then append that code to the password (no spaces or other separating characters), and then submit. That should help you get passed the time limit.