Even so, that's just using a cart and a hired one at that. I mean the analogy of a horse to a car (as in something you owned personally and rode around on) was never really true, at least in cities. Also, speaking of London, the Underground is surprisingly old -- the first parts of the system opened in 1863. Yes, Sherlock Holmes is often depicted as going by hackney, but had he really existed, he could have ridden the Tube.
Until 1976 the law was still extant that they had to keep a bale of hay in the vehicle.
The prices were even regulated
http://www.londonancestor.com/stow/stow-hack.htm
Indeed the Romans even had regulations about road width to ensure drawn carts could pass each other
https://en.m.wikipedia.org/wiki/Roman_roads