They are usually self-employed
That varies wildly depending on city, county, state, and (I’m guessing) country requirements. I drove for a few years in a small-ish city, and neither me nor anyone I ran into at the usual spots were self-employed. We all worked for one of two cab companies, making hourly wages plus tip.
No, you’re describing a co-op. Which my town does have, right across the street from the for-profit grocery store.