Yes, in many (most?) cities in the US, you need a special license to spontaneously pick up passengers (e.g. when you wave a taxi over to come pick you up). However, town-car services, where you pre-arrange to be taken somewhere ahead of time operate under looser rules. Uber's loophole was making an app so you technically prearranged to be picked up a few minutes before you were picked up.
Can't speak for the US, but in a UK city it wasn't exactly difficult to get the equivalent by ringing up at short notice (or indeed get one straight away by turning up at their city centre office, or spotting one illegally touting for business on the streets at pub closing time). Often they were cheap too. Even a phoneline which became an app attempting to aggregate the firms existed (I knew the guy who ran it... which is probably why I knew it existed!). All Uber brought to the party was VC money, the social expectation to award every driver 5 out of 5 and more aggressive surge pricing
