First, we should acknowledge that it is difficult to generalize about either native Australian or American peoples, who after all lived in a variety of different environments and developed very different cultures. Southeastern Indians, for instance, had far different lifestyles that were far different from those who lived on the Great Plains. But there are, perhaps, some grounds for comparison.
First, most aboriginal Australians practiced hunting and gathering, which contributed to a very mobile society with a fairly egalitarian (equal) social structure. Not all Native Americans lived this way, but many did, at least at the time of European contact. The Micmacs, for instance, who lived in what is now Nova Scotia, collected shellfish, lobsters, fish, and sea mammals, a lifestyle which had much in common with people who lived along the southern coast of Australia and in Tasmania. By way of contrast, many Native Americans lived in highly stratified societies. A person who lived in Cahokia, a city of up to 30,000 people, or any of the Aztec or Maya ceremonial centers would not have recognized many aspects of Australian life. No such societies existed in Australia, due to a lack of agriculture.
Another similarity, at least for some Indian peoples, would be religion. Many Native Americans, including some who lived in settled agricultural societies, held a view of religion that can be described as animist, much like Australians. They believed that spirits were immanent in nature, rather than inhabiting a separate realm from people. In this, both societies were quite different than Eurasian faiths, which tended to differentiate between the spiritual and the secular. People in the Americas and Australia saw dreams, in particular, as an important aspect of religious experience.
Finally, there are similarities between the political organizations of many aboriginal groups in both Australia and the Americas, especially North America. Aboriginal political units, called moeties by anthropologists, were organized along kinship lines, with members of clans interacting through commerce, warfare, and especially marriage. Native Americans were also organized along clan lines, even in some of the more complex societies. Interestingly, some aboriginal and Native American societies were matrilineal, meaning that property and status passed through the maternal line rather than the paternal (father's) line. Among southeastern Indians, this practice meant that a child's maternal uncle, rather than a father, would play a central role in raising him or her. Not all Indians or Australians adhered to this practice, however.
One more similarity, a modern one, should be noted: Both Aborigines and Native Americans were violently dispossessed of their lands, and subjected to concerted attempts to destroy their culture and force them to conform to western society. Both peoples continue to struggle to maintain old cultural forms as well as lands, especially sacred sites to this day.
Source: Michael Leroy Oberg, Native America: A History (John Wiley & Sons, 2010)