Countries like Australia and New Zealand are usually classified to be part of Australasia. At times, the island of New Guinea is also included plus other neighboring islands. However, Australia and New Zealand are two countries usually considered to be part of Australasia. Australasia is just a subset of Oceania i.e., a sub-region under Oceania.
Generally, Oceania covers all the islands that are in Australasia, and other sub-regions like Micronesia, Polynesia, and Melanesia. The fact that Australia and New Zealand are grouped together to be called Australasia does not mean they belong to the same continent. Australia is a continent, the second smallest continent.
While New Zealand does not really belong to any continent, and that is why it is most times considered to be under Oceania. The only thing that made them be grouped together as Australasia is the fact that both belong to a region under Oceania known as Australasia.