Answer:
Like other western countries, cats and dogs are probably indeed the more common house pet. Nationwide you can find pet shops with birds, numerous types of fish, also guinea pigs are a very popular house pet.
Other pets that I know of numerous Australian house holds having are rats, mice, snakes (the carpet snake kind so non-venomous) lizards and turtles.
As Australia has so much outback and farm land, there are plenty of pets in the form of horses. I'm from rural Australia and have even known friends on property to have pet goats.