Apple TV

You are probably exceeding the time limit; some have reported that its is merely 30 seconds between receiving the verification code and submitting.

Enter your password, but do not submit yet. Request a (new) code, and append that to the password, without space or other separator character. Then submit. Entering 6 digits should be doable within the given time.

Is your Apple TV 3.9 inch and black, or 7.7 inch and silver-white? The black models are 2nd generation or later.

Identify your Apple TV model - Apple Support

If you have two-factor authentication for your Apple ID, and a 2nd or 3rd generation with older software, then the sign-in procedure is a bit special:

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.

There is a time limit before the code expires, so request your code (or a new code*) after entering the characters of your password, to use those precious seconds effectively for just the code.