Is it possible for a person/company to find out how many bank accounts
someone has? (and which bank(s) they are with?)

- it doesnt really matter what type of bank account or the account
number, just how many accounts they have and with who. 

Im looking for a yes/no answer and a little information on roughly how
somone can find out information about a persons bank accounts.
(especially in Australia)

As an individual it would be very difficult to get this information. 
However, the tax authorities can by way of the security no/personal
account no. etc. as applicable to each countries.  There is no
instrument in public domain which would give you this information.

That is, it is an emphatic NO to your question.