I would hazard a guess that this is a false equivalency. Sydney and Melbourne probably started out small and as they enlarged into a huge metropolitan conurbation, only the original part maintained the administrative designation "City of X". The City of London, while making up part of the ancient city of London, usually refers to The City of London Corporation which was a financial center made (again) independent from royal authority during the reign of William III to allow its banks to charge interest rates.

From the book Where Does Money Come From (2011): The Dutch brought the concept to England following the successful invasion by the Dutch Prince William III of Orange in the ‘Glorious Revolution’ of 1688, who displaced the reigning monarch. Under the reign of William III (and Mary), the charging of interest (usury) was soon allowed and bank-friendly laws were introduced. Given the previous repeated defaults by indebted kings and a raid on the national mint; Parliament and creditors of the state (namely the merchants and goldsmiths of the Corporation of London) lobbied for the creation of a privately owned Bank with public privileges – the Bank of England – and the secession of one square mile of central London as a quasi-sovereign state within the state.

Sydney is the same (as are I suspect most of the other Australian major cities).

Melbourne residents love comparing themselves to London.

Also Buenos Aires and City of Buenos Aires.