I don't think that's true of Australia. American Australians tend to assimilate pretty well, and after a while people tend to mostly forget they were Americans originally, even if they still have a bit of an accent. The Australian politician Kristina Keneally is an example. She was born in Nevada, grew up in Ohio, didn't move to Australia until her 20s. But I don't think anyone really thinks of her as "an American". She's an Australian politician. You might like her politics or you might dislike them, but nobody really cares about where she was born and grew up. She's one of us now.
The Australian media has even taken to (at times) calling Virginia Roberts Giuffre (the most notable public victim of Jeffrey Epstein) "an Australian", without qualification. (She married an Australian man, had kids with him, now they live here.) Whereas the American media just calls her an American.
The Australian media always wanted to view Mel Gibson as an Australian, even when he said that he himself identified as an American rather than as an Australian. (I think they are less keen on that now that he has made himself a bit of a persona non grata through his behaviour. America, you can have him.)