Apple TV gen 1
If your Apple TV is a 3.9" black box, then it is at least 2nd generation.
Identify your Apple TV model - Apple Support
"An Apple ID verification code is required to sign in. Type your password followed by the verification code shown on your other devices".
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.