Is my Apple TV too old

I have an older Apple TV. Model MC572LL/A. The copywright/TM is showing 2007.

Apple TV (2nd generation) (A1378/MC572LL/A) is a 2010 model (sold from 2010 to 2012), so that discrepancy is a mystery.

Identify your Apple TV model - Apple Support

Maybe it's too old to do that [add apps]??

Indeed. You’ll need a 4th generation or later model to be able to add apps from the App Store.

Every time I [try to log in], I get an error that says, 'Apple ID or Password or incorrect'.

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.

That is a second generation model. The following applies to it as well as the 3rd generation.

See the following from Download apps on your Apple TV - Apple Support

The App Store isn’t available on Apple TV (3rd generation). If you have this model, you can't download new apps, but you can update your software to update your built-in apps.