It occurred to me that I might be wrong about the locally stored database, so I conducted an experiment.
I put my phone in airplane mode, then went to my record player and played a song with my phone sitting nearby. Within a minute, my phone correctly displayed the currently playing song, despite having no connectivity whatsoever. This proves that there is a local database of songs against which the service can compare what it hears. Obviously the database does not include every song ever written, that would be ridiculous.
I never claimed that the phone was not listening, it has to listen in some way to recognize music. What I did claim, and have now proven, is that it can identify songs without sending the audio to Google.
IIRC, it uses a database of common and popular songs stored locally on your phone (possibly adapted to what Google knows about your taste in music, idk) and only goes online for matches when you do a manual song search.