Your mortgage lender will have a copy of the loan policy on file. They do not receive a copy of the owner policy, so if you need it to file a claim, the loan policy won't help you. If you have lost your title policy and need to make a claim, ask for a copy of the HUD-1 Settlement Statement which will prove that you paid for an owner policy.
If you are looking for a copy of any title policy just to get a discount on a new policy, the loan policy should suffice. Some lenders will provide a copy readily upon request, other lenders won't. Again, you may be able to use a copy of the HUD-1 Settlement Statement as proof of coverage.