Most international destinations are governed by bilateral treaties under which specific carriers are designated to fly specific routes between the two countries. For Australia, those carriers were originally Pan Am (route authority bought by United in 1985) and Continental (route authority dormant). UA took advantage of the dormant CO authority to add SFO-SYD flight, which they previously did not have the rights to.

For AA to get rights to fly to Australia, they would need the government to reopen negotiations with Australia to create additional frequencies and to allow additional carriers to be designated.

Same issue in Japan and most of the rest of the world. Europe is different in that many countries have adopted open skies agreements with the USA.