It is clear that something has gone astray somewhere.
Your first contact needs to be with the clamping company, pay and get your car released.
Yes it's a pain, and possibly wrong, but neither of those get you access to your vehicle.
Your second contact needs to be the DVLA, and you can discuss your payments to them, and ask for receipt of them, if indeed they have received them.
Once you have unravelled the situation, if you can prove that clamping company were not in receipt of information to lead them to legitimately clamp your car, you can sue them.
If they do have legit info, then hopefully you can claim your fees back.