Drive not rent
You're thinking of RelayRides, which will rent your car from you when it's not being used. (You're correct, that does not sound good for an Alfa.)
Uber turns your car and you into a taxi service. With the Uber app, the passenger and you link up and you receive a percent of the fare. You drive.
Does Uber really let you select the car you want to be picked up in? Hey, maybe it should.