Yes, an entity (Individual or Company) can register several business names using the same ABN.
In fact, if you intend to trade under a new name, you must register it as a business name to avoid penalties.
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).
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Updated — Mar 13, 2019