The government office could try to assert copyright, but it would be an uphill battle.

As a matter of public policy “the U.S. Copyright Office will not register a government edict that has been issued by any state, local, or territorial government, including legislative enactments, judicial decision, administrative rulings, public ordinances, or similar types of official legal materials.” U.S. Copyright Office Practices § 313.6©(2).

https://garson-law.com/can-state-governments-own-rights-in-copyright/

A mugshot isn’t a law, so maybe this doesn’t apply cleanly, but the copyright office clearly isn’t rushing to hand out rulings in favor of publicly created works.