Even before installing the app on my phone, I appear to be displayed as “Away on mobile” on both the Battle.net desktop application and Battle.net mobile app.
In these cases, I was not logged in on any device at all (and no instances of Battle.net.exe were running on my PC either). I have never used any third-party apps or services.
This problem persists even after changing my password (and thus, for sure, not being logged in anywhere at all).