Although there isn't a defining date when the Inhabitants of Australia (for the purpose of this answer, Australians will refer to the White members of society with British heritage) developed their own national identity, the 'ANZAC spirit' which is shared with New Zealand, can be seen to show widespread national identity of what it meant to be an Australian rather than a British citizen living on the Australian continent.

The ANZAC spirit came about as a result of the Gallipoli campaign during the First World War, fourteen years after the federalisation of Australia, in which Australian soldiers were deployed as Australians (in contrast to the boer war, where they were deployed as British soldiers).

The Australian troops, being from a more rural part of the world that their British counterparts, were generally seen to be fitter and more enduring than British troops. The hardships endured by the ANZACS at Gallipoli, along with the casualties suffered, led to a feeling back home of misunderstanding, and in some cases even anger at why Australians were fighting and dying in a war thousands of miles away from their homes.

This ANZAC spirit eventually led to a feeling of national character, and the virtues displayed by the troops defined what it meant to be Australian, this is a view that is still held by many today, and wheeled out by politicians from time to time. This was also coupled with the fact that many Australians were now native born, rather than having emigrated from Britain, like many of their fathers in the previous century, took pride in the land that they lived in. However, rather than seeing themselves as an independent Australian, they were Australian-British citizens within a global empire.

This answer was off the top of my head during my lunch hour, but hopefully this has answered your question. Follow up questions are more than welcome!