When the app is started you should see a system tray symbol in the lower left corner. But for PureOS Gnome version I don’t see such a menu. There is only the App Menu next to activities (see my first post). I don’t know if this is PureOS specific or if this is a general Gnome issue.
Anyway when you install the Gnome Shell Extension ("(K)StatusNotifierItem/AppIndicator Support"). This restores an older behaviour of system tray for Gnome.
Now you see an additional tray menu in the the upper right corner and with a right mouse click you will finally see the menu to initialize your nitro key.
@smojo: Thank you so much for this detailed explanation and your effort fo even include pictures and everything!
And yes, I’m sure this will definitely help a lot of people here!