Uber is getting pretty cheap too, regardless. Example: my wife just caught a car to SFO from Berkeley for less than $80, and no tip. That's cheaper than a taxi, friendlier, you can call 10 minutes before you need the Uber, and track it to your door.
A taxi would need to be reserved well in advance, would show up at an indeterminate time, the driver would be indeterminately sleepy, and you'd have to deal with tipping and credit cards. He might be smoking too, and if any of this happened with Uber, you'd leave a bad review. A few reviews, no more driver.
Of course, a person on a budget is going to take BART or a shuttle.
Of course, in some cities the cabs don't go the places they are technically "required" to go -- they give a very long estimate of arrival as a way of getting you to go away. Or don't show up at all. I suspect this varies a lot by geographical location.
The worst cab situation in SF is trying to find a cab at night, which Uber has greatly improved upon.
Buses and trains, in many cities, don't go everywhere, and don't always run when you need to go, so, yes, poor people take cabs.
I have no problem at all with Uber's black car service, but I think the taxi service is sketchy.