It turned out that using two-factor authentication on the Apple TV 2nd or 3rd generation doesn’t work naturally. There is no way (field) to enter the additional code and no information on how to use this. There is just an error message stating that your username, password combination isn’t correct.
The solution to using two-factor authentication on Apple TV 2nd generation
It took me a while to find it and I actually needed help from Apple Support on Twitter to find a solution. The solution is almost on the bottom of Apple Support on using Two-factor Authentication.
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.
Given that there is an expiration on the verification code that didn’t work well for me using the Apple Remote. So, the tip is to use the Remote on an iPad or an iPhone. I even used my MacBook Pro to generate the verification code and then swiftly enter it after the password using the Remote on my iPad.