SAN FRANCISCO—The judge overseeing litigation targeting Facebook Inc. with claims related to its Cambridge Analytica scandal had a series of tough questions for plaintiffs lawyers pitching to lead the potentially precedent-setting case.

U.S. District Judge Vince Chhabria of the Northern District of California, who was assigned the batch of class actions against Facebook and the now-defunct data analytics firm in June by the Judicial Panel on Multidistrict Litigation, questioned how Facebook users were harmed when Cambridge Analytica accessed data without permission. The judge also puzzled over how the litigation would proceed since Cambridge Analytica filed for bankruptcy, staying civil litigation against it.