Yes, business names are registered to an ABN and multiple business names can be registered to it. If you intend to trade under a particular name you must register it.
It is an offence under section 18 of the Business Names Registration Act 2011 (Cth) to carry on a business under an unregistered business name. Failure to adhere to this provision will result in a penalty of 30 penalty units ($3300).
This information is of a general nature only and does not constitute professional advice. You must seek professional advice in relation to your particular circumstances before acting.
Updated — Jun 8, 2018