The UK has a similar distinction. "Hackney carriages" (the classic "black cab") are allowed to pick-up passengers on the street, while private hire vehicles (often called "minicabs") can only be used by pre-booking. The former are more regulated than the latter. Ubers are minicabs.
(Black cabs don't have to be black, but usually are. As to why they're called "Hackney carriages" - the last person to know the reason probably died in 1863.)
In the US at least, there are two classifications for "car for hire."
One is street-hail - you wave down a car or get into one at a stand. That was heavily regulated and taxi companies had the relevant licenses.
The other is "town car". You call for a town-car and it shows up. town-car was very lightly regulated.
Yes, every taxi company offered town-car services, but there were lots of town car companies that didn't do street-hail.
Uber/lyft are town-car companies. Neither one does street-hail.