no, most are independent contractors who lease the cab and the medallion from the taxi company. If you have your own medallion, you lease that to the cab company (so you are making money from it even when you are not working, which you can't do 24/7). Cabbies have always been independent contractors with very few exceptions.

The cab company leases the medallions and has some store of medallions it owns. It buys and maintains the cars and runs the dispatching service.

I am from Brazil, here taxi drivers usually own the medallion and the cab...

The medallions is suppposed to be nontransferrable and free, you would only need to pass some certification tests, but there are ridiculous levels of corruption, and a pinch of violence here and there (For example drivers in certain spots violently banning others from using the same spots, even with guns if they think is needed).

Here Taxi Mafia runs strong, but is still made of individuals, not companies.