Indigenous people would not be treated much differently than under the British. Murder, dispossessment, and genocide unfortunately :-( If France considers the colony part of the metropole as opposed to a separate 'dominion', it might make it harder for surviving Aboriginal communities to fight for land rights in modern times.
On the other hand, France may send less colonists out than OTL's British did, assuming they keep OTL's pattern of not being as deeply into settler colonies. This will be good for the Aborigines. While Aboriginal culture might suffer due to missionary activity and the power of the Catholic Church, the Church in at least some cases could prove to be a strong defender of the lives and rights of converted Aborigines. Rather than trying to forcibly assimilate mixed-race children like OTL's British Australia, French law may instead recognize an Australian version of gens de couler libre.
Settlement-Just as in New Caledonia and Devil's Island, Australia will be a place for the French to dump convicts/political prisoners. In Fact, Australia may replace those places as the main destination of French convicts-which will probably be much better for the convicts in question. The nice Mediterranean climate of southwestern Australia or the temperate climate of the Southeast and Tasmania would probably be much more healthful for them than the tropical hell of Devil's Island. If French colonial settlement is lighter than the British, they may permanently import a coolie population from China or India that will result in a large French Australian Asian minority.
Resources-For this I don't really know enough to even speculate, beyond suggesting that lavender may become a common cash crop in the southwest, imported from Provence. Resource extraction by prison or coolie labor is possible. And wine, of course, will probably become an Australian industry earlier than IOTL.
Political Structure-As I said above, assuming it follows the pattern of rule over OTL's colony, les Champs d'Elysee will keep a pretty tight leash on French Australia, treating it as just another part of France that happens to be hard to get to, like rural Brittany. Perhaps a history of sending political prisoners will result in more extreme politics being practiced in French Australia, with various radicals and freethinkers cooking up new movements among the settlers in a land where the authorities are interested more in keeping them out of the way than under control.