I bought a car and accidentally, without reading, clicked one of my favorite cars in the game, it then said I couldn't go to fix my car because it belongs to someone else. I then removed myself from the vehicle and when I tried getting back into it, it said: "you don't have access to this personal vehicle".
When you purchased the new car, and selected your old car, you replaced the old car with the new one. You no longer own the old car, so have no access to it. To get it back, your only option is to buy it again.
Edit: It has come to my attention that you may still have the lost car, and it is simply in storage; apparently you can actually store any vehicle in storage if, for instance, you trade in a 10-car garage property for a 2-car; it will automatically store the 8 extra vehicles, and they can be retrieved at a later date by using the vehicle management menu in a garage or by calling your mechanic in-game. Check there to see if you still own the lost vehicle.
You could, technically, contact rockstar support, and they might be able to get your old car back, but they dont do that often.