Nor is Scotland or Northern Ireland, but they're talking about independence.
Scotland has been its own traditional country for a very long time, and the union with England was formed when the King of Scotland became also King of England and not the other way around.
Northern Ireland is the vestige of a brutal occupation of Ireland by England. The Catholic population traditionally loyal to Ireland now outnumbers the UK-supporting Protestants.
London has never been a country. This is obviously a protest demonstration and not a serious proposition.
Whilst the argument is for all of London to be independent. It would surely be possible for the 'City of London' to remain in the EU.