"Paid all the taxes when we picked it up from the dealer." You actually aren't paying taxes on the gain at this point - all you are doing is setting aside dollars to be applied to your tax at the end of the year when you do your return.
This is because no one knows at the point when you pick the car up how much tax you will owe on this car. The tax you owe is your margin tax rate times the net value of the prize. I don't imagine that this was discussed when you picked up the car.
For example, the W-2G might have had a standard 20% withheld on the prize. However, if you are in the 22% tax bracket, then (within limits), you would owe at the end of the year an additional 2% of the value of the car (22%-20%). Of course, you might not have been that close.
See how that works?
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