Terrorist Organization Classification Act of 2023 This bill directs the Department of State to designate five specified drug cartels as foreign terrorist organizations. (Among other things, such a designation allows the Department of the Treasury to require financial institutions to block transactions involving the organization.) The five specified cartels in the bill are MS-13, Queen Nation, Sinaloa, Cartel Jalisco Nueva Generacion, and Beltran Leyva Organization. The bill also requires the State Department to submit a detailed report on those five cartels and any other cartels it may identify. Based on this report, the State Department must designate as a foreign terrorist organization any such identified cartel (or faction thereof) that meets certain criteria for designation as a foreign terrorist organization. The bill specifies that it may not be construed to expand eligibility for asylum.