National Geographic has a nice webpage, that explains:

There are six major kinds of islands: continental (1), tidal (2), barrier (3), oceanic (4), coral (5), and artificial (6).

The article continues that when Pangaea broke up, "some large chunks of land split. These fragments of land became islands. Greenland and Madagascar are these type of continental islands."

Referring to an image (below) of Pangaea (source: Amante, C. and Eakins, B. W. 2009), it is clear that Australia is it's own continental mass, while Greenland is a piece of a larger mass. Despite Australia historically being referred to as an island country "because of its lack of land borders", it is not an island. There is a nice Wikipedia article that explains:

Europeans discovered Australia in 1606, but for some time it was taken as part of Asia. By the late 18th century, some geographers considered it a continent in its own right, making it the sixth (or fifth for those still taking America as a single continent). In 1813, Samuel Butler wrote of Australia as "New Holland, an immense island, which some geographers dignify with the appellation of another continent" and the Oxford English Dictionary was just as equivocal some decades later. It was in the 1950s that the concept of Oceania as a "great division" of the world was replaced by the concept of Australia as a continent.

It's also important to consider that sea level has a major role in how much continental mass is "island". If you consider land bridges like the Thule Bridge, one could postulate that Greenland has not always been an "island".