Australia does get widely referred to as an island as well as as a continent - an island-continent sometimes. My understanding is that "continent" in terms of geology refers to the largest land mass of a continental plate and is more a matter of convention - a general agreement - than precise definition. Such definitions have been subject to revision.
Australia can be seen as a very "continenty" continent. It contains very old stable continental lithosphere in cratons, like for instance the Pilbara or the Yilgarn craton.
I fear, the quiz makers were correct with classifying Australia as a continent.