A partial answer... StatusNotifier is not enabled by default in MATE.
Install "dconf-editor", run it to set org>mate>panel>general>enable-sni-support
calibre will then run in the background (with "calibre --start-in-tray") but in my case I then get an error popup saying, "Notification Area" has quit unexpectedly. If you reload a panel object, it will automatically be added back to the panel."
The network connection/speaker etc. icons are then not showing. If I click "Reload" then they reappear. There is no indication there that calibre is running, but it is actually properly running in the background and works as server.
More:
https://github.com/mate-desktop/mate-desktop/issues/286
https://github.com/mate-desktop/mate-desktop/issues/288
If would be helpful if others could give their experience with this.