Take it up with Wikipedia. like someone else pointed out there is not really a consensus as to what is specifically called a continent or even how many continents there are.
Should you find it weird when any Australian calls themselves Australian and not Commonwealth of Australian? Because as you can see that's the full name.
I'm from Colombia and thought it was weird at first since in Spanish you refer to Americans as Estado Unidenses. Which in hindsight is the weird thing because you don't really do that for other countries. You don't call people from la Republica de Colombia "Republicanos" you call them "Colombianos".
If people from the United States of America want to call themselves Americans, then it makes 100% sense since the "America" in the name of the country is the only part that's actually a proper name and not a descriptor.
It's not both though. America isn't a country, and neither is Oceania. America is the name of the continent, and a nickname for the US, and Oceania incoporates several countries, including New Zealand and Papua New Guinea
True that continents are defined differently in different parts of the world. In Europe, there's America and Oceania as continents. In "America", America is a country, and so is Africa :p
(jokes aside, it's true that in the USA continents are defined differently than Europe, for example. There's North America & South America as separate continents, and yes, Australia as a continent).