It sounds like you may have an Apple TV (2nd or 3rd generation). Identify your Apple TV model - Apple Support
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.
The Apple TV+ subscription is compatible with Apple TV (3rd generation or later) with up-to-date system software. Unfortunately Apple TV (2nd generation) is not supported for this streaming service.
Being subscribed is why you are asked to sign in. Being subscribed or not isn’t related to the endless loop, but due to two-factor authentication on an older device, and not doing the sign-in as described above.