Australia is very large and very remote from most of the rest of the world.
Both of those things may seem kind of obvious, but they had profound implications for how Australia developed.
For thousands of years, Aborigine populations were basically isolated from the rest of the world. We're not sure exactly how they got there in the first place, but what we do know is that they had very little contact with anyone else once they did. As a result their language and culture is very distinctive.
Fast forward to the early 17th century, when Dutch explorers found this huge island continent nobody had any records of. But no European settlements were established there for over a century; it was simply too far away.
Then in the late 18th century, Great Britain decided that this huge island continent would be a great place to leave the criminals they didn't know what else to do with. The fact that Australia was huge meant that they could easily fit as many people as necessary. The fact that it was remote meant that the criminals would have basically no chance of making it back to Great Britain on their own.
But when you put a bunch of people together for a long time, even if they were initially all criminals, they tend to start setting down roots and establishing towns that grow into cities and eventually building whole nations. Once this started happening in Australia, more migrants came from Britain, voluntarily for a change. Their remoteness from Britain gave them a significant amount of autonomy in developing their own political system, even though they remain to this day officially under the Crown of the United Kingdom.
Because most of Australia is desert, the British population became concentrated on the fertile east coast, especially the southeast coast. In the sad refrain of colonialism throughout history, when the British settlers found land they liked they claimed it, regardless of what the indigenous Aborigines thought about the matter. Fertile land was relatively scarce, and often fought over. (On the other hand, the abundance of fertile land in the United States didn't seem to stop conflicts from arising between colonists and Native Americans.)
Most of Australia's political institutions were borrowed from Britain, but some were also modeled on the United States, which gained independence around the time Australia started being colonized. One way their geography may have influenced these decisions is that travel from Australia to the US is actually faster than travel from Australia to the UK, because crossing the Pacific, while difficult, is still easier than crossing all the way around Asia and Africa up to Europe. (This was particularly true until the construction of the Suez Canal and then the invention of the airplane.)
As a result of this hybridization Australia has a parliamentary system, but with a Senate and House of Representatives like the US instead of a House of Commons and House of Lords like the UK. They have a Prime Minister rather than a President. Their highest court has 7 justices, compared to 9 in the US or 12 in the UK.
Australia's proximity to Southeast Asia has also influenced their cultural development, and there has been substantial trade and immigration between Australia and many Asian countries. In the 20th century this process greatly expanded, particularly due to the rapid economic development of Asian countries such as Japan and Korea. Today, there are substantial cultural and social influences from Asia in Australia, and Australia is tightly integrated into Asian trade networks.