As documented in the Tray API docs, the click event is ignored for tray icons that have been created as an app indicator, i.e on Ubuntu Unity.
This means "menu bar" style applications do not work on Ubuntu (see max-mapper/menubar#1), making them not really cross-platform.

The reason there is no click event being exposed is that these kind of applications do not fit the "Ubuntu status menu design guidelines" and therefor libappindicator does not support it.

The proper solution would be to have Ubuntu alter there style guides and make the app indicator adhere the Status Notifier specification, or at least the Activate method. This has already been requested in Unity libappindicator bug 522152, which hasn't received much attention since it's creation in 2010, regardless of it's high bug-heat score.
Tl;dr: I don't think this will happen any time soon.

My plan is to work around the limitations set by libappindicator by listening directly to the signals sent by the Unity Panel service as described by @sjakthol. I already created a PoC.