some vehicles with engines need registration (the license is for the driver).
quads, motorbkes and karts need registration, probably, where you live.
on some roads, some additional restrictions apply, indirectly. minimum speed on the highway is 50kph so if a vehicle is not able to go that fast is (indirectly) not allowed to be on the highway. also most highway codes do not allow bicycles etc.
the initial idea of yours "roads are for cars and motorbikes" is, at best, inaccurate.
there is law, however, in almost every country, that dictates where you may and where you may not go with a bicycle.
i doubt in australia you are forbidden to go with a bicycle on the road.