Oh I see.
Australia most definitely allows double citizenship and as far as I know you do need to be a citizen to be counted as domestic. Having citizenship in essence means you can get a passport BUT you don't have to get one. I think the piece of paper saying you're a citizen is what is required...
That is correct. Historically, Australia would expect you to drop your other citizenship on becoming Australian citizen, unless your original country does not allow you to renounce it (for example, France, and until 1990, Poland). In those cases you were allowed to keep the old citizenship in parallel (have dual citizenship).
At the same time though, until recently (2000?) you would lose the Australian citizenship if you subsequently accepted a new citizenship. This caused some unfortunate incidents, for example famous Australian architect Harry Seidler (1923-2006) whose birth country, Austria, restored his Austrian citizenship that he lost as a child refugee after the Nazi takeover. There was a huge public outcry when subsequently his Australian citizenship was quietly cancelled and he could not renew his passport despite holding it since 1958. Following this debacle, Australia changed the law and does not automatically cancel its citizenship when people acquire a new one (which is why Andrew Durante was allowed to take up NZ citizenship and not lose his Aussie one).