To register a .au domain name various criteria has to be met.
The requirements vary depending on what extension you arre aiming to register.
For example the requirements for .com.au are quite different to .org.au
To be very general for a .com.au or .net.au usually an ABN or ACN will suffice however usage criteria or business name matching is also requested.
The requirements for a .org.au are quite different and more stringent.
The Australian domain authority auDA have a tool that you can use to quickly find the list of requirements for any .au domain name.