This “eye” icon opens a popup menu containing the favorites currently displayed in the Customize window. The favorites do not need to be saved to be viewed in this menu. It can be used as a preview of the menu you are editing.
This popup menu can also be used to see the favorites found in the Search window. If some favorites in the search result have the same name, duplicates name in the popup menu will have the [!] suffix in order to make each name temporarily unique.