While vehicles are parked on a dealer's lot, they are actually financed by a financial institution of some sort. The dealership, therefore, is required to have the vehicles readilly available for routine audits that are conducted by the financial institution. Once the car disappears from the dealer's lot, the financial institution knows that it should be paid for that car immediately.
I know this because, as a teen, I worked in the business office of a dealership, and know how stringent and "nosey" these audits can be.
Knowing that this is the case, the dealership is likely trying to make it appear that the car has not been sold yet, therefore they will not have to pay the financial institution. They want the car in possession when the auditor shows up.
The bottom line is that I am suspecting fraud on the part of the dealership. What I cannot fathom is why they would tell you that they had to do an inventory on the car.
Was this a new car or a used car? Does the title and paperwork you have on the car match the VIN on the car itself?