well of course there were horse drawn carriages and even just jumping on someone’s camel or horse or whatever else was probably a thing. but before driving as a paid service, i doubt it was called a taxi or anything like we call it today or since the invention of the automotive. 1635 was the first horse-drawn carriage as a paid service in england at least. without doing more research i’d assume you would just ask to ride with someone and offer money as compensation. something we see today as how taxi service is known, but hardly the oldest concept in civilization when civilization spans back thousands upon thousands of years before the last 500 or even thousand years we are talking about right now.
what i’m saying is that the concept of how a taxi functions, yes no refuting it’s existence for as long as civilizations and something to ride has existed. but the word or an equatable older word that transitioned into the modern word of taxi or cab? not even close to as old as civilization existence.