It has nothing to do with Firefox. Firefox has no idea what other applications are installed and would not be able to provide the file path of the respective apps (as denoted by the folder icon on the right).

Under what conditions does this search popup appear?

Well, I went from a narrow keyboard to a full-sized keyboard and after about a week, I think I got one of these pop-ups. But they only seem to happen in FF. However, that's probably where I type the most. I guess I need to do more hunting. Thanks