Hi there,
In the 1890s all of the six self governing colonies that later became Australia would have been responsible for their own regulations re records of outward bound passengers. But, do you know which colony he was in, as perhaps there would be outward passenger lists to check to give clues for possible ship. Deaths should be noted in the ship's log, and also reported at the next port of call for that ship.
Have you checked marine deaths at sea as many ships calling into ports in Australasia were British registered
http://www.nationalarchives.gov.uk/help-with-your-research/research-guides/birth-marriage-death-sea-or-abroad/If his death was onboard a ship that was still in Australian waters heading to another Australia port prior to sailing further, you may need to look at the various registers for each of those six former colonies. There is no central register for Australia. Each state/territory continues to have their own parliaments and the institutions established in the 1800s continue to be governed by each jurisdictions own laws.
Here's RChat's Australian Resource Board :
http://www.rootschat.com/forum/australia-resources-offers/ Within that are child boards for each State/territory and live links (where available) for their respective BDMS.
Here's Victoria's outward bound departures
http://prov.vic.gov.au/provguide-50 NSW did not regularly require outward bound lists until about 1898.
Cheers, JM