Much of the special status of London was granted before 1189, and it retains its special because of time immemorial concept and English common law.
I won’t bore you with all the details but there’s loads of weird stuff like a mayor that only lasts a year, companies get the vote based on number of employees, separate police for from the rest of London etc etc. That’s barely scratching the surface.
Can’t easily be changed because some of the “rights and liberties” predates written common law and are “senior”. Of course, when push comes to shove they find a way but that rarely happens.