Does a person renting someone else's car have a right to anonymity?
I'm not taking a stand on fingerprints per se, but identification is something that does matter. Specifically: my right to be free from having someone else masquerade as me. There's the rub...
No. The customer is voluntarily entering a contract with a private party. If the customer doesn't like the terms of the rental agreement, they can decide not to enter the contract.
The car's owner has private property rights that allow him to make requirements of customers as he sees fit. If the car owner requires more than the market is willing to bear, whether it be high prices or fingerprints, the market will respond negatively and the car owner may be pressured by economic concerns to reconsider the terms of his contract.
The only rights relevant to this case are private property rights, and the right to voluntarily enter or decline a contract.