Unable to sign in to Apple TV 2nd generation

If you have Two-Factor Authentication active, then you will need to append the verification code to the end of the password. The verification code will expire after a short while, appearing as not accepting the password.

Re: appleTV keeps asking for appleID, password and authentication code circularly

Two-factor authentication for Apple ID - Apple Support

“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.”

Did you also try the advise in the first link? First enter the characters of your password, then request a code, then enter the code in the 30 seconds before the code expires.