I called them and they said they were not liable. How so? The ID is not counterfeit, it's from the DMV.
The federal and state governments possess what the law calls "sovereign immunity." This the principle, carried over from the law regarding the British crown, the government is immune from liability for money claims asserted against it except where it has expressly waived that immunity. The federal and state governments have a variety of statutes that waive immunity for certain types of harm. But these statutes are very specific and are strictly construed in the state's favor. Thus, your first hurdle here is identifying a CA statute or CA state constitutional provision that waives the state's immunity for claims regarding DMV license screw-ups.
If you can find that, your next hurdle here is proving that the state was negligenct and thus liable to you. Not all mistakes give rise to negligence claims. The issue will be whether the state had taken reasonable steps to ensure its system worked properly. If it did, then it is not liable, even though an occasional mistake gets made.
Finally, there is the issue of damages. You are only entitled to recover for the actual financial harm caused by this. Aside from your name (if your signature is legible) and the photo, the other person got no other infomation about you from the DMV. Your name and photo are not generally protected information to begin with. What has this other person done with the information that has harmed you financially?