A few thoughts,
In my experience, it is rare, very rare, for a company to ask for a DL and driver's license information. I have provided a photocopy of the former once or twice to companies but not the latter. What I have seen more often are questions asking if I have a driver's license and auto insurance. Those questions I routinely answered without providing specific information.
I have difficulty understanding why a company that uses independent contractors would need that information unless their lawyers fear that for that company we really aren't independent contractors. I won't go into the discussion about what makes someone an employee or independent contractor. There is plenty of discussion of that on the forum or you can look at the IRS site for the factors. What I will say is the industry norm is not to require that information because we are not employees. If the specific job the OP cites is not the norm, maybe that is an exception.
LisaSTL states, "When considering starting another small business I was going to be required to have proof of car insurance and a rather large liability policy for one specific client. In the beginning I was just going to be a sole proprietor with no employees. IMO it was excessive, but on the other hand we sometimes have to do what our clients request." I ask what about that job made it necessary? A plumber needs insurance in case they screw up doing their job, the homeowner can fix the damages using money from the insurance. What can mystery shoppers screw up that auto insurance can fix?
What would I do? I would be very reluctant to give that information. It is a big deal to give out personal information to those that don't need it. Then it gets passed around to others who also don't need it.
Payment - I think I would have responded something like - What did your lawyers say about paying a shopper for work completed according to the contract? How would they respond to a lawsuit? The correct answer would be the shopper needs to get paid for work completed according to the contract. Additional information can be collected from shoppers before any future shops.
Mystery Mojo - If you want to get more from the forum members you should name the company and perhaps even the type of shop (without also naming the client). There are probably lots of other shoppers who work for that company and some who did that shop. Only if you share can others offer their experiences.
Mystery Mojo - payment, or the lack thereof is a really big deal with many of us. The forum is the only place shoppers can share information about companies that don't pay. If you don't know about Freeman, I suggest you read the threads on that company. Many of us want to know about any non-paying company so that we can avoid them before they go into bankruptcy or worse. Please share the company name and any followup information about your situation.
Happily shopping Rhode Island and nearby Massachusetts and Connecticut