New Zealand is part of a continental mass called Zealandia. It’s considered a ‘micro-continent’ which broke off from the Australian plate and is now submerged underwater.
There is confusion where New Zealand fits because it’s also grouped as part of Oceania, which is a geographical region rather than a continent.
Oceania includes New Zealand, comprises Australia, New Guinea and the many Pacific islands. Oceania isn’t a single tectonic plate, and it’s comprised of many which would make it easy to consider Oceania a continent.
New Zealand is on the border of the Pacific, and Indo-Austral tectonic plates, meaning that is not part of Australia or Asia.