Tray icons are not available in current gnome version (or you need some extensions). I just added a "Stop" button to the GUI: https://github.com/deveee/kazam/commit/b10dead0bbc623c801d2f892b390919b466b971a
Maybe I will add it as an option in settings window and then I will send a patch. Because latest stable version is from 2014 and I don't think that it will be fixed upstream. And without a possibility to stop recording, the application is not really usable.
