This issue appears to be exclusive to the Spotify app -- system sounds, the basic Samsung music app and YouTube all properly adjust the volume. I have tried two clean installs with no success, and have taken care to also remove / disable all equalisers that might impact Spotify.
I'm running the app on a Samsung Galaxy Note 3 (SM-N9005) with a custom ROM port of the Note 7's 6.0.1 Android firmware. I have a premium account and the app works perfectly fine on my PCs and my old HTC One.
Solved! Go to Solution.