That seems incorrect. The mayor is the mayor of Greater London which includes the City of London, although the City also has a Lord Mayor, as London boroughs have elected or ceremonial mayors. In other words, the City has its own mayor in the sense that other parts of London have their own sub-governments, although with a few differences such as policing and city status.

> The mayor is scrutinised by the London Assembly and, supported by their Mayoral Cabinet, directs the entirety of London, including the City of London (for which there is also the Lord Mayor of the City of London)

https://en.wikipedia.org/wiki/Mayor_of_London

> In 1965 the county was abolished and replaced by Greater London, a two-tier administrative area governed by the Greater London Council, thirty-two London boroughs, and the City of London Corporation.

https://en.m.wikipedia.org/wiki/Greater_London