That'd be like me and you being in a secret club with a secret handshake. We've been meeting at the same bar every week for years. All of sudden, your twin brother shows up one day and, as always, I go to do our secret handshake and he can't do it. He looks like you. He comes from the same parents. I have no reason to suspect that it's not you because "you" showed up just like you do every week. But I know something's not right because he doesn't know the handshake.
It's the same thing with an iPhone. That's exactly how you know that it's not third-parties that Apple cares about and that they're not trying to actively prevent anything done by third parties. If that's all it was, an OEM Apple sensor would be fine. It's not, though, because it has nothing to do with third-party vs. first-party and everything to do with security.
An interruption in connection, or a change in device identifier on either end is cause to require an “apple-authorized” reset, or discarding/ignoring any sensitive personally identifiable information (PII), of which fingerprints are definitely PII.