Australia basically only has one city per state with nothing for hundreds of kilometres in any direction so everything close to the city center is just considered part of Melbourne. With “City of Melbourne” being responsible for a small section of it.
You wouldn’t use “city of Melbourne” in conversation for any reason other than to talk about the council of that area.
Sydney (and to a lesser extent the other capitals) is massive: I think last time I looked it's the third the size of the Netherlands. There are a number of councils/LGAs in Sydney, one of which is the City of Sydney (https://en.wikipedia.org/wiki/City_of_Sydney). There is/was an attempt to reduce that number, for various reasons. A similar attempt was done in Queensland, which did make the City of Brisbane LGA (https://en.wikipedia.org/wiki/City_of_Brisbane) basically the same as "Brisbane".