ASX use a system called CHESS (Clearing House Electronic Subregister System) to track ownership and facilitate settlement of securities traded on the ASX.
When you create an account with a broker, you are allocated a 'Holder Identification Number' or 'HIN'. This number identifies you as a CHESS sponsored shareholder. You should keep your HIN private in the same way as you would a bank account number, as it can be used to personally identify you and access your holdings.
You can have multiple holdings under a single HIN, and even have multiple HINs under one name. You can identify a HIN as it will begin with the letter 'X' followed by 10 numbers (e.g. X0001234567).