It depends on what you mean by the term, as there are different senses of "taxi service". In many cases, the law has long differentiated (for non-flaky reasons) between taxi services and limo services[1]. A taxi service is allowed to pick up street hails (i.e. arranged on the spot, by line-of-sight), while a limo service had to be booked and paid for through a third party.

In a way, that makes regulatory sense -- there are many issues that arise with taxis (in this sense) that don't with limos -- unnamed randos getting into a stranger's car, large amounts of cash floating around, clogging up pickup points, fighting for fares, dubious upcharges. In that sense, Uber has functioned more as a limo service than a taxi service (except for this historic association between limo services and expensive cars).

OTOH, I agree that it's ridiculous for them to claim it's "not a taxi service" in the sense of "just a technology company, bro!". It's certainly a ride service of some kind, for which safety/insurance regs are applicable and reasonable (or at least, not outrageous).

(Ditto for the claim of "we just match people up!" -- they obviously are functioning more as a service provider with tight control of the product than some kind of marketplace or matchmaker.)

[1] Despite it's name, that doesn't necessarily mean "stretch limo"; it could be a sedan.