Facial Recognition and Biometric Technology Moratorium Act of 2021 This bill imposes limits on the use of biometric surveillance systems, such as facial recognition systems, by federal and state government entities. A federal agency or official may not in an official capacity acquire, possess, or use in the United States any such system or information obtained by such a system unless Congress passes an act that specifically authorizes such a use. Such an act of Congress must contain certain provisions, such as provisions naming the specific authorized entity and auditing requirements relating to the system. Information obtained in violation of this bill shall not be admissible by the federal government in any proceeding or investigation, except in a proceeding alleging a violation of this bill. An individual aggrieved by a violation of these restrictions shall have the right to sue. Any state officer authorized to sue on behalf of the state's residents shall also have the right to sue on behalf of the state's aggrieved residents. A state or local government unit shall not receive certain federal law enforcement grants unless the government unit complies with a law or policy that is substantially similar to this bill's restrictions on acquiring and using biometric surveillance systems.