Yes, Apple's apps are exempted from the Microphone Privacy settings.
Shazam is Apple-owned, however, the Shazam app still follows third-party app rules, it seems - microphone permission is required.
There's also an optional "Music Recognition" control for the Control Center. It uses the Shazam icon (and is undoubtedly powered by Shazam), but it is a built-in function of the OS (unlike the Shazam app). If you delete the Shazam app, the Control Center's "Music Recognition" control is still present.
So, the way I interpret things is this:
Apple acquired Shazam and built it into Siri ("Hey Siri, what's this song?").
Apple created a Control Center shortcut to access that same functionality.
If you still download and use the Shazam app for its extended features, it's still being treated as a third-party app despite Apple's ownership.