Without hearing the whole exchange, it's difficult to know whether the taxi driver had also behaved badly. As I understand it, black cab drivers own or rent their cabs and are self-employed, so can record a conversation if they want to - perhaps for security reasons?
It's not so much that David Mellor lost his temper - I suppose most people have "lost it" and said stupid things at some time in their lives - it was his attitude that comes across as particularly unpleasant. He was in effect telling the taxi driver that he was a little nobody whose only role in life was to take orders from his "betters". I doubt that this attitude is exclusive to David Mellor but the Sun and other tabloid newspapers have always despised him and I wonder if they'd have been so keen to print this story if it had been someone more to their liking.
It did make me smile a bit to think that DM's partner is Chairman of Visit England, this incident not being a great advertisement for doing so.