It's not particularly independent, but it is the last of the unreformed medieval municipal corporations. In earlier times every borough/city corporation was chartered ad hoc and often had completely undemocratic rules for membership (e.g. selection by the outgoing members, maintaining a self-selected oligarchy, or election by an extremely restricted franchise). There was a process from the Municipal Corporations Act 1835[0] onwards where the old order was reformed into a more modern (more uniform and more democratic) county/district/borough/city councils system. We eventually ended up with the local authorities we have today.

Except for the City of London. Here, a combination of the Victorian power of the Corporation, coupled with the tiny resident population of the City (and a consequent desire by central governments to leave elections in the effective hands of the companies who employ people in the City rather than those who live there) have left a body allowed to take its own direction. Along with the Council of the Isles of Scilly (weird because Scilly is far too small to justify its own principal authority, but too far from Cornwall to be conveniently governed by it) it is one of only two really sui generis principal local authorities in England.

[0]: https://en.wikipedia.org/wiki/Municipal_Corporations_Act_183...