What is so sacrosanct about a taxi? The cabbie wasn't recording a private conversation that his passengers were having - that would obviously be an invasion of privacy. He recorded an exchange between himself and David Mellor. If such abusive language had occurred in a different place of work, I would imagine a customer/client would be asked to leave.
Without hearing the whole of the conversation, there is always a doubt that something else occurred or was said that incited Mellor's behaviour but, even if that were the case, his attitude and attempt to humiliate the driver was dreadful.
Whether some cabbies are annoyed about the driver recording this exchange is, I think, irrelevant. No working person should be subjected to this sort of ignorant behaviour without there being some kind of comeback - otherwise such people will think they can behave as they please.