Financial Institution Customer Protection Act of 2023 This bill specifies that a federal banking agency may not request or order a depository institution to terminate a customer account unless (1) the agency has a valid reason for doing so, and (2) that reason is not based solely on reputation risk. Valid reasons for terminating an account include threats to national security and involvement in terrorist financing, including state sponsorship of terrorism. A federal banking agency requesting a termination must provide the depository institution with notification and justification. The bill also sets forth additional requirements for the Department of Justice when seeking subpoenas, summoning witnesses, or compelling document production in the course of conducting a civil investigation in contemplation of a civil proceeding involving certain banking laws.