If I have registered as an Uber driver, can I drive a car that is not registered in my name?

The car has to be registered (and approved) on YOUR driver Uber account.  I believe there may be a way that a bonafide company that hires drivers can operate with the same vehicle for multiple users,  but that takes it beyond the personal driver accounts that we're all using.  You would have to contact Uber to see how to do that if it's even possible these days.

And with the rates as they are these days, I can't fathom anyone hiring drivers any longer as the driver pay hardly covers driver costs without allowing any profit for the auto owner IMO, but at one time it was possible when the rates were much higher. 

For a personal driver account, the car has to meet qualifications, be registered AND insured in YOUR name (or your name and someone else's such as your significant other) to be approved.

My son drove for Uber 2 months ago for a few weeks as he was between jobs . The car is in his name, the insurance is in his name, but the registration is in my name. He was able to drive for Uber. Maybe a mistake on Uber's part but he was approved to drive.