How are passwords stored by Apple TV?
The Apple TV (2nd and 3rd generations) runs iOS under the hood, so it almost certainly (hard to be 100% sure without doing some substantial hacking) uses the system keychain, just like iOS on an iPhone or iPad.

I defer to someone with a more concrete knowledge of the iOS security layers, but my understanding is that like most other things on iOS, keychain access is sandboxed, so an app can only access the data that belongs to it. There are further security mechanisms in the keychain, but I don't know enough to speak about them with confidence. Based on the answers to this StackOverflow question, I think that it is possible to gain access to keychain data if you have physical access to the Apple TV, but it's not a trivial task.