Oceania is a useful geographic region, but I haven’t seen anyone include it as a continent; the most common definitions I’ve seen for continents are “lands sitting on the same tectonic plate” (so, the geological definition) and “contiguous land of a sufficient size not broken up by any ocean.” Interestingly, both of those definitions still allow some wiggle room in what counts and what doesn’t, but in either definition, Australia is not part of the South Pacific islands.