The authenticator, wherever you installed and registered it, produces the OTP code. That's all it does.*
The authenticator just produces a code, based on a shared secret with turnstile.
It does not matter where it is installed.
It does not matter how you enter the code (whether it's copy and paste on the same computer, or manual entry to a different one).
There is only one code (which changes every 30 seconds, that's just how it works) based on the shared secret when the authenticator is registered with the system. There is no knowledge, or data link, shared between the authenticator and the requesting system. Only the six digit code (derived from a key shared between the authenticator and the authentication requesting system when the authenticator is first registered). So it doesn't matter where the authenticator is, as long as you can retrieve the current code when you are logging in elsewhere.
If you register a new authenticator, then the old authenticator will no longer produce the correct code, because you have changed the key share that generates valid codes. You don't want to do that.**
If you want to have the authenticator accessible elsewhere, you need to copy the initial key, or backup and restore the authenticator keys....depending on what methods your specific authenticator supports.
*This is all explained in this article.
**You do want to do that if you no longer have access to the authenticator that you registered. Just click "Lost your device?" instead of entering the OTP and register a new authenticator. You will need access to your registered email to follow the workflow.