I figured out that Zotero 7 uses some default icon in a tab of the mate-panel. I manipulated zotero.desktop in /usr/share/applications/ to point to /usr/lib/zotero/icons/icon32.png, but it does not affect it.
If I look on apps, whose icon is displayed I can see, that their settings are straightforward, e.g. Icon = sublime_text and then there are multiple locations which differ probably by file resolution like: ~/Icon/48x48/sublime-text.png.
So how does Ubuntu parse this? Does it mean I need to set icon directories by resolution and they have to be in the same directory as exec?