By that definition, no cab companies are cabs, since that is exactly what the cab companies do--arrange trips between independently contracted cab drivers and potential customers.
Under the legal definition of cab services in many municipalities, Uber is a cab service rather than a livery service. In other municipalities, Uber is merely a livery service. In the U.S., each municipality gets to define what a cab service and a livery service are (because that is a local power under the federal system of government).
It's a communications network that links private car services with people who want to get a ride from the private car service.