Many years ago, dealerships insisted that we fill out a loan application no matter how strenuously we told them we wouldn't be getting dealer financing. That seems to have changed over the years. It isn't just them looking at me as an experienced buyer, my children have been able to buy cars without dealer financing and without having to complete a loan application.
I wonder if they are able to do a soft pull of your credit score based just off your drivers license info. Remember they scanned your license before you took a test drive. That soft pull would give them a ballpark estimate of how good you are with credit, though there would be no income info, or info regarding existing debt levels.
Once you complete the sale the state may require them to collect other information to complete the title and registration process. This could include information not on your drivers license.
If you are selling a car they will need to have proof you own the old car, before buying it from you.
Neither of these things is needed during the negotiating phase.
If I was asked to complete a loan application I would refuse. They want an application so that they can add the loan details to the negotiating phase. If they know the range of loans they can use it makes their job easier, and yours harder.