My family attorney sent my final custody case files, including all personal info (tax and bank records) to another client?

You wouldn't likely recover anything unless there are provable damages....as in the disclosure of the information led to fraud in your name. You can report the attorney to the NM bar, but it was a mistake. If that other person is another client, they probably contacted the attorney, and sent the records back or destroyed them. They wouldn't want there stuff sent to another client and have that info be misused. If the attorney let you know that a mistake was made, then they have likely taken the actions to correct their mistake and they are honest enough to tell you.

Everybody makes mistakes. However, your family attorney made a serious blunder in sending your personal financial data -- which presumably contained your Social Security and bank account numbers -- to a third party. Good attorneys want to have satisfied clients who will recommend their services to others. Write a polite letter to your attorney expressing your dissatisfaction with how your private financial information was mishandled by his firm. Express your concern that complete strangers now have access to your personal financial information and Social Security number. Suggest that he either make an adjustment to your bill or that he provide you with one year of identity theft protection coverage. And if the attorney does not provide the identity theft protection coverage, I suggest you buy it yourself for your own peace of mind.