This paper proposes the Digital Identity Rights Framework (DIRF) to address threats such as digital cloning, sophisticated impersonation, and unauthorized commercialization of identity-related data arising from the advancement of generative artificial intelligence (AI). DIRF is a structural security and governance model comprised of nine domains and 63 control elements designed to protect digital similarity attributes based on behavior, biometrics, and personality. It integrates legal, technical, and hybrid enforcement mechanisms to ensure consent, traceability, and monetization of digital identities. This paper presents the DIRF's architectural foundation, enforcement strategies, and key use cases, providing platform builders, legal authorities, and regulators with the essential controls needed to strengthen identity rights in AI-based systems.