The privacy concern is not about fingerprints. Unless you wear gloves all the time, you are leaving fingerprints everywhere, all the time.
The concern is about corporations and government logging and saving everything you do on the Internet and in the real world. Sometimes all they save is the metadata, such as who you called or texted, our what time you drove past a camera that could read your license plate, and sometimes they save the content of your conversations or emails. They do this to everyone, even people who don't think they are important.
As we become more and more digital, the trails we leave behind will become more numerous. We need laws limiting what the companies and government can do with that data. Would you like to get a ticket every time you exceed the speed limit? It could be done automatically, and you really did break the law, so you'll deserve it, you lawbreaker, you.
In the past, some of these automatic tickets (like red light cameras) have been challenged on the basis that they don't know if it is you behind the wheel. But if your car has a fingerprint reader so it only starts when you touch it (a nice feature that will dissuade carjacking), then they do know it was you. If you make a phone call while you are in the car, they know it was you. They might even have a voice print to use against you in court if you decide to challenge the ticket. And you don't even need to be an enemy of the state to be a target. Just an ordinary person with a cell phone and a car.