Yes.


State
 name of relevant act/special conditions and requirements
 ACT
Yes for dogs, not for cats. A dog must be registered to a keeper if it is over the age of 56 days, has changed keeper for 28 days, or is owned by a person who has been a resident for 28 days. Domestic Animals Act 2000.
 NSW
Yes. A dog and cat must be registered to a keeper if it is over the age of 6 months. Companion Animals Act 1998.
 NT
 No.
 QLD
Yes for dogs. A dog over the age of 12 weeks must be registered.  For cats - in some council areas. Animal Management (Cats and Dogs) Act 2008.
 SA
Yes for dogs.  A dog of 3 months age must be registered. For cats - in some council areas. Dog and Cat Management Act 1995.
 TAS
Only for dogs. A dog over the age of 6 months must be registered. Dog Control Act 2000.
 VIC
Yes. A dog and cat must be registered if it is over 3 months old. Domestic Animals Act 1994.
 WA
Yes. A dog 3 months or older must be registered. Dog Act 1976. A cat over six months must be registered. Cat Act 2011.