Not for me. Mine only allows Google. Maybe because I'm using a custom default search engine? If so, it should just use my custom one, and should obviously give me a choice regardless.
Either way, to reiterate, it's definitely Firefox you're seeing here. If you don't like the way it behaves, I recommend following the other suggestions on this thread to file a bug report with Mozilla.
Change your default search engine to something else and back again, and you'll notice that the widget updates.