Apple TV

You can go into settings - about to see which model number it is

Those older models don’t have the App Store, and only have preset “channels” that are sent via server.

If you had a 2nd gen then it is classified as obsolete and no longer supported. The 3rd Gen has a server version of the TV app, but won’t be getting any other new content as it can’t run apps that most providers are now using, existing providers are also starting to drop support for their server versions.

When using in on an older device using 2 factor authentication you need to enter the password and code together. However, in order to avoid the time limit you would need to enter the password, then manually request the code, and then add it to the password

Get a verification code and sign in with two-factor authentication - Apple Support

The Apple TV app would be preinstalled on 4th gen and 4K Apple TV models. If you have a 3rd gen or earlier model then it doesn't have access to the app store. However, the 3rd gen has a server version of the TV app, make sure it is up to date.

You can verify your model here

Identify your Apple TV model - Apple Support