If you are talking about independence, no. The US won independence through a violent rebellion. Australia became independent through a very gradual and consensual process, so gradual it is hard to pinpoint the exact moment at which Australia became independent.
A major step was the Statute of Westminster in 1931, by which the British Empire extended full legal sovereignty to its "Dominions". The Canadian, Irish and South African governments were keen on this law, so it applied to them immediately. However, the Australians, New Zealanders and Newfoundlanders (at the time not yet part of Canada) were hesitant, so the UK agreed the law granting them sovereignty would not apply until their own national parliaments consented to it. Australia didn't agree to accept it until 1942, New Zealand didn't until 1947, and Newfoundland never did (which helped pave the way for its annexation by Canada).
So Americans fight a bloody war for their independence, Australians are offered (a big chunk of it) by the UK and drag their feet for over a decade before accepting the offer, and New Zealanders for even longer.
For example, the cheesecake box says "Sara Lee, est. 1971 Lisarow NSW", except it was already a 50 year old American company when they expanded to Australia in 1971.
Or the "made in Australia from at least 12% Australian ingredients" labels on ALL the food. Like https://content.api.news/v3/images/bin/5b8b48cb5d8051c3c6734...
Yesterday I picked up a bottle of mustard that said American Mustard, proudly made in Australia. It's surreal!
Let's not even mention the pavlova.