Military and Veteran Caregiver Student Loan Relief Act of 2023 This bill makes designated primary family caregivers of veterans eligible for student loan forgiveness under the Public Service Loan Forgiveness (PSLF) program. Under the PSLF program, the Department of Education must cancel the balance of interest and principal due on a borrower's Federal Direct Loans after the borrower makes 120 monthly loan payments while employed in a public service job. This bill broadens the definition of public service job to include service as a primary family caregiver of an eligible veteran participating in the Program of Comprehensive Assistance for Family Caregivers.