I finally found a way to use Shazam without the app, as I usually listen music on my PC with headphones, and holding my phone against them was awkward. And I can always do with less tracking.
SongRec (https://github.com/marin-m/SongRec, #Linux only) works marvelously. It can listen directly to your speaker output and identified the song immediately.
It still uses Shazam's services for identification, but it only gives them a fingerprint of the audio (and your IP, I guess)