Australia is counted as continent in all generally used landmass continent divination systems but three continent divination (Afro-Eurasia, America, Antarctica).
Commonwealth of Australia, Australia as short has got it name from Continent that it locates, same way as United States of America is sometimes called America, although acronym USA is more popular.
Sometimes term Oceania is used to denote Australia continent, even though it is officially name for region that locates about in same area, although both Oceania and Australia (continent) have regions that other don't have. In technical content other terms to used to denote Australia continent are Sahul, Australinea or Meganesia (from wikipedia), when there is need to distinguish it from the Australian mainland.
Although in questionnaires and even generally speaking term Oceania is preferred as Micronesia and Polynesia aren't part of any continent (in continent divination systems that are based on landmass continents).
This is the first time I have seen the term "Australia" being used.