You likely have claims against both the background check company AND the employer under the Fair Credit Reporting Act (FCRA):
Against the CRA (background check company): FCRA ยง 1681e(b) requires them to follow "reasonable procedures to assure maximum possible accuracy." Mixed-file cases (reporting someone else's records as yours) are a common FCRA violation. If they didn't match on SSN, that's a strong accuracy claim.
Against the employer: FCRA ยง 1681b(b)(3) requires the employer to provide you with a copy of the report and a "reasonable period" to dispute it BEFORE taking adverse action. If they rescinded the offer without giving you a chance to dispute, that's a separate violation.
Damages can include: lost wages, emotional distress, and attorney fees. If the violation was willful, statutory damages of $100-$1,000 per violation plus punitive damages. Most FCRA attorneys work on contingency. This is a strong case.