This is a great idea. Not only can the police verify that a given photo ID matches the person in front of them, they can also verify that the ID is valid and unaltered by verifying that the details on the ID match the details in the DMV's database, eliminating fake IDs from being an issue. This wouldn't be 100% perfect -- maybe a really determined ID thief could get the DMV to issue them an ID in someone else's name -- but it would dramatically increase the risk and makes ID theft much harder to scale.
A federal effort to standardize an identity verification service across federal and local offices nationwide would be helpful. The service should be available to any entity (not only banks or financial entities) who wishes to verify the identity of a counterparty. The process and fee should be standardized nationwide, with the fee being break-even and paid by the entity requesting the verification.
Post offices are a good candidate to offer such a service, but would need some work to set up (unlike police agencies, I presume post offices don't have access to DMV databases).