Is there a way to sync Shazam songs identified through the Mac Control Center with Apple Music?

If I Shazam a song by opening the app directly on my iPhone or Android, it gets added to the Shazam playlist in Apple Music. However, if I use the Control Center Shazam button on Mac, the recognized song doesn’t get added to Apple Music, nor does it appear in the Shazam app itself, which is quite annoying.

For example, if I have 250 songs recognized through the Control Center, I’d have to manually search and add each one to Apple Music. It’s also frustrating that I have to open the Control Center history and add songs manually, instead of having them sync automatically.

Basically, the Control Center Shazam feature and the Shazam app still work in isolation, there’s no sync between them. Apple should really just add a “Listen” widget from the app itself at this point, because the Control Center version feels pointless if it only recognizes music without syncing to the Shazam app or Apple Music.