Apple Music: 'My Library' vs 'Mac Mini Library'?

The default location for the Apple Music app library is: /Macintosh HD/Users/yourusername/Music/Music Library.musiclibrary (or Home/Music/Music Library.musiclibrary)

The difference in the two libraries that you note is unknown to us because neither of those library file names is standard. Perhaps if you open each and examine the music within you can answer that question for yourself.

To select an alternative music library, press and hold the Option key when launching the Music app.

https://support.apple.com/guide/music/use-multiple-libraries-mus7663a920a/mac

This appears to me to indicate that Mini Library is a shared library that is available as an accessible option for the Music app. Is there another Mac at your location? It may be your Mac's own library being made available for other computer in the household.

I don't know where that library resides and I don't see that option on my Macs. I haven't yet upgraded to Tahoe, so I suppose it may be a feature of the Music app under that OS. Or I may not see it simply because I have not set my music library to be shared.

You may find additional info here: Share songs from a shared media library in Music on Mac - Apple Support

You're welcome.