Is it though?

Yes, that is why people prefer it because it is a different experience. Some people still prefer hailing a cab.

It is someone (or some company) that you pay to have a driver take you from place A to place B.

Yes that is why Uber and a taxi company are comparable, but point is saying "Uber is a taxi company" is nonsense. They are not, they are similar, they are not the same.

Does it really matter how they get the information?

For the customer, no. For the structure of company/service, yes, which is the topic here.

Isn't that pretty much the definition of running a taxi cab service?

That is the experience you get from a taxi cab service, but that still doesn't make Uber a taxi company. That makes them comparable, they are, as a business, far different. Thus saying "They own no vehicles!" is nonsense.

Let's say you order from an online service that allows you to order from ALL burger places in your area. You still order food, you still end up with a burger, you still pay the money for a burger. Is that online service a burger place or a restaurant itself? No. They are just a party between burger places and you.

Same with Uber, they are a connection between you and a bunch of "freelance" drivers that are connected to Uber. But none of the vehicles are owned by Uber and their job is not to own cars. Same as online burger ordering service's job not being cooking burgers.

So saying "They are a taxi company" is false, they are not a taxi company, they have nothing to do with taxis. They have nothing to do with owning vehicles.