Hello @kiroma, what are the contents of ~/.steam/ubuntu12_32/steam-runtime/pinned_libs_*? If you completely close steam, then move or remove the pinned_libs folders, then restart steam, does the steam tray icon behave?
There's no ~./.steam/ubuntu* but there is ~./.steam/steam/ubuntu*, so here you go.
~/.steam/steam/ubuntu12_32/steam-runtime$ ls ./pinned_libs_*
./pinned_libs_32:
./pinned_libs_64:
has_pins libGLU.so.1 system_libdbusmenu-gtk.so.4
libdbusmenu-glib.so.4 libpangox-1.0.so.0 system_libGLU.so.1
libdbusmenu-gtk.so.4 system_libdbusmenu-glib.so.4 system_libpangox-1.0.so.0
After deleting the directories it works fine.
This has fixed my tray icon not working as well.
This is how it looked before:

This issue started happening to me after updating my Linux Mint from 19.2 to 19.3
Nothing extracted yet.
Your system information
Please describe your issue in as much detail as possible:
At random when I try to open a menu via right clicking the steam icon in tray, the menu doesn't open, and it produces an error in console:
(steam:9031): LIBDBUSMENU-GLIB-WARNING **: About to Show called on an item wihtout submenus. We're ignoring it.Steps for reproducing this issue:
Unfortunately, I haven't got any reliable steps to reproduce the issue.