No, the distinctive thing is that you're paying for a ride.
It seems to me that the distinctive thing is that this is pre-arranged and negotiated with a specific driver.
If I hail a taxi on the street, I don't know the driver, and I don't want to haggle over the fare. So it is reasonable for the government to regulate that, and set standard fares.
If I arrange for a ride over the internet, I can independently check out the driver's reputation and negotiate the fare in the comfort of my home. So there is no need for government involvement.
On a few occasions I have met taxi drivers that were especially helpful and friendly, and I negotiated special "meter-off" trips over the next few days. My family and I had a great tour of London a few years ago. Nobody knows London like a cabbie.