I have taken a look into the eddie-ui folder under path /usr/lib/eddie-ui, and see that there is a file there called eddie-tray.
This is perhaps a file developed by AirVPN that is supposed to let Eddie minimise to tray?
The libappindicator.so.1, that on previous versions of Debian took care of this, is not possible to install on Debian 11.
A user should not have to create hacks to solve problems related to a commercial app. I think AirVPN should be able to fix this for their users.
For now I am leaving the app icon on the panel. Hopefully AirVPN would be able to fix this little problem in a later update to Eddie.
Edit:
Minimising Eddie to tray worked fine on Debian 10 with Eddie 2.20.
It also worked on previous versions of Eddie on Debian 9.