It depends who/what is the appointed trustee of the discretionary trust. Assets held in trust are held in the name of the trustee. If you personally are the trustee, then usually/ideally you would open some sort of sub-account that says something like "Joe Bloggs, as Trustee for the Bloggs Family Trust". If the trustee is a company, then the company must open a trading account in its name. something like "Company XYX Pty Ltd, as Trustee for the Bloggs Family Trust". That makes it 100% clear which assets are held in trust and which are held by you personally.
PS: I *strongly* suggest you have a chat with your accountant and/or solicitor to get proper advice here, as stuffing it up could be a very expensive mistake to rectify (due to capital gains tax event/issues etc), and a lot depends on the specifics of your set-up.