But humor aside, where do you draw the line? Nearly all of the Mojave desert in the US was clearly a sea floor based on the fossils found in its sandstone rocks. But we're not going to call it an ocean. It was an ocean, and then things moved around and now it is a desert. Zealandia was a continent but now its the seafloor around an island.
Lets say we call it a continent, does it suddenly make it all the territory of the government of New Zealand? Does that mean they control fishing and mineral rights for hundreds of kilometers beyond the shores of their island? Where does that lead?
Probably, yes. France extended its EEZ last year with new evidence that the land around Kerguelen island forms an underwater "continent" as well. I imagine if this finding about Zealandia gets confirmed, both New Zealand and France (from New Caledonia) will want to extend their EEZ there as well.
Note, the extension is from 200mi (the eez) up to 350mi -- so NZ wouldn't get the entire continental area.. and in that extended area, they only get exclusive rights to the soil and subsoil (so things like oil, etc... but no exclusive fishing rights).
There are economic and strategic assets here, whether it is oil or methane clathrates, fisheries, or places to establish forward operating bases, these things are 'easy' to claim in your 'own' territory and 'hard' to claim in international waters. Step 1, redefine them to not be international waters. See the South China Sea and now Zealandia ...
 Extended Economic Zone
Totally understand the implications of EEZ, but at some point you have to draw the line else the line will become increasingly more arbitrary as countries try to fuzz the boundaries of what a continent is and is not. Why does any country get to lay claim on a chunk of seafloor that has been uninhabitable for land animals far longer than humans have been around as a species?
United Nations Convention on the Law of the Sea
It's true this is unfair to marine animals, but that's really their fault for not attending the meetings when the treaty was signed.
I wonder if it'd be possible to build a sealab on one of them; it looks like there are often small atolls nearby, and it could be a good environment for testing concepts related to building and maintaining livable underwater environments.
Things quickly get challenging with depth.
I have pity on the Chinese national geographic toponym bureau for having the tedious task of translating this for national linguistic unity.
Truly, we dwell in a post-national mire of bureaucrats! We need more solid continents!
But France is still in the picture since New Caledonia also sits on the plateau of Zealandia... New Caledonia itself named after Scotland!
Also, just because continents are touching, does not mean they aren't separate. Europe, Asia, and Africa are all one continuous land mass. (The Suez canal technically separated Africa from Asia, but it certainly doesn't cut through the depth of the entire continent.) And we're only one Bering Strait away from connecting all that to North and South America.
In this case, the light rocks are no vey tick, so the thin layer of water that we call ocean can cover them. I like this image: https://goo.gl/images/aqCBJL