I think that's because "Oz" comes from "Aussie", which Americans mispronounce (they say something like "ossy" rather than "ozzy").
It's uncommon for Australians to say "Oz" in my experience, but it's still definitely a thing (albeit mostly in ads/marketing rather than daily speech). And even the ones who never say it still know what it means.
I don't know what people say in the UK because I have never been there.