No. The laws underpinning motor vehicle registrations can tend to be quite complex in understanding, requiring reference to multiple sections and legislative instruments to decipher.
However, basically put it runs like this:
- The government makes laws about the use of roads and vehicles in general. These laws are generally stated to apply to everyone (including government);
- The law defines what classes of moving objects require registration, which we'll call a 'motor vehicle';
- Any 'motor vehicle' which is driven or towed on a government regulated road must be registered with the relevant Australian government authority for the jurisdiction relating to the car. This is usually determined by its 'garage address' (or similar), but there are reciprocal rules allowing a vehicle registered in one state to be driven in another; and
- Driving or towing an unregistered vehicle on a road is an offence.
There are usually exceptions to all of the above requirements, but they are very specific in scope. For example, a new vehicle which is not yet registered may be driven on a road under specific circumstances for test driving.
There's no scope in the above for anyone to set up a competing system of registration. It would not be legally recognised, and all vehicles 'registered' under it would be considered unregistered for the purposes of the law.