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.