The car must be in your name by law for registration, insurance, and inspections. Your wife needs to have your name on the cards to operate the motor vehicle.
If you choose to rent a car through Uber, you can log on to your account via uber.com, then find the rent with Uber.
It sounds like you’re asking if you can drive on your wife’s Uber account. Absolutely not! She went through the application and vetting process and it is her name and face on the account.
Any passenger would rightly be alarmed to be picked up by a man when Uber showed them a woman driver is on her way. They will report you, Uber will shut down her account, and neither of you will be able to drive. There could even be a scenario where an alarmed passenger may call the police.