Language - we use some pretty unique language and idioms. You may not consider them very Australian because you use them all the time but they are. The combination of those makes it Australian.
Landscape - also the light and colours of Australia are very different from the dullness of Europe, greens of England or the colours of USA. Also, beaches and that lifestyle is very unique to Australia and the outback.
Architecture - the way buildings are set up (we have laundries, in England they don't), taps, light switches, roads, etc
Historical context - references to events and world history in an Australian context is important. WWI means something very different to an Aussie, that it does to a Yank (to us, it's the defining of what it means to be Aussie - victory in a defeat, and all that).
Cultures - Australia reflects a different culture from ALL other English speaking countries. We are also becoming a greater mix of nationalities, which influences the food, smells, dress, look and so on of everyone.
Then there is mateship, the laid back - she'll be right - attitude and so on. Very Aussie.
To completely understand what is very Aussie, you need to read more overseas books, watch movies that are very Aussie and very not and, if you can, travel overseas to see it.
But I hope that helps a little. But try googling "What it means to be Australian". I did and came up with these:
What it means to be Australian... Melbourne Trivia Reference Melbourne Guide
What it Means to be an Australian
I'm an Australian
Here is a guide for writing in an Indigienous Australian context that may help you further understand what it may mean:
http://www.australiacouncil.gov.au/__data/assets/pdf_file/0018/32373/Writing_protocol_guide.pdf
