The “Australian dream” is a longstanding marker that has always been used to describe the so-called successful Australian person.
An Australian is said to live the “Australian dream” or the “Great Australian Dream” if they own a house with a backyard in the suburbs. Such a person who lives this dream is deemed a respected person. A respected person who most likely speaks with the Australian accent. A respected person who most likely grew up in a predominantly Anglo suburb and went to public school. A respected person who is a middle-aged Caucasian adult holding a stable job in a company run by high-paying Caucasian executives so as to support the family and pay off the mortgage.
There’s seems to be an air of “whiteness” associated with the “Australian dream” and that’s why I don’t favour the phrase too much.