Thank you for referencing my list! In case anyone wants some extra info: The frontends for Audire and Audile are open source. Audire uses the proprietary Shazam API, but Audile uses the open source APIs AudD and ACRCloud.
Note that the clients being FOSS is of little relevance because all they do is forward a recording to a blackbox proprietary service run by a for-profit company.
The code that has access to your audio and does the actual task at hand is not FOSS in the slightest.
There's Ambient Music Mod, "Port of Now Playing from Pixels to other Android devices". It's not the same, but it allows for your device to recognise popular music even while offline, possibly on a regular basis. It requires Shizuku, though.
Audile is on F-droid, though it uses AudD for the actual music recognition backend. I'm not sure it's possible to have a FOSS backend for this kind of service.