There's multiple, sometimes overlapping ways of altering the DPI globally on Linux. You should first ensure these are properly configured and Spotify is still not using them correctly.
However, Spotify provides a
--force-device-scale-factor argument for controlling DPI scaling.
First, test it by running
spotify --force-device-scale-factor=2.5 in a terminal, it should run with all elements much bigger. Then tweak the number to figure out what value you like.
Second, find the
.desktop file for this and modify the command. Often it may be in a path like
/usr/share/applications/spotify.desktop. Instead of modifying this system-level file, you may want to create a user override for it in your home directory.