Is there a company like Uber in either Rome or Florence?

I've found that taxis in European cities are easy to use and fairly well regulated, so I have no hesitation using them as a 60ish woman traveling by myself.

No. The type of car share service provided by Uber or Lyft was banned by the courts in Italy in 2015.
The court ruled that only licensed professional drivers can provide that service.

You can use taxicabs or private limo service.

Uber operates the Uber Black service in Rome and Milan. It works with an app like in America, but it is a black limo service. With Uber Black the driver picking you up is not Joe Doe driving his family Toyota, but rather a licensed professional black limo driver driving a limo company luxury Mercedes Benz or similar vehicle. The price is therefore higher than a regular cab.