its to late to get it now. Basically what they would need to do is inspect the car for any issues (this includes your current one) That way they can see what state the vehicle is in. Then the contract will NOT go into effect until 30 days after the fact. It will not cover any pre-existing issues either noted on the inspection either.
Also if I'm not mistaken, the previous owner has one month to transfer his extended warranty to the new owner. By now I'm assuming its to late.
It would also be identity theft/fraud to act like your the previous owner. The dealership should never have given you the information of the previous owner either. They would be liable and could be heavily fined for processing the claim.