Connect to my account

Bough used Apple TV 2 gen

The 2010 model is quite limited in 2021. I hope you didn’t pay much and don’t expect much beyond iTunes Store and Netflix.

and can’t connect it to my iTunes account

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.

You wouldn’t have to do that often, if the credentials don’t change.