Recently, Artificial Intelligence (AI)-driven Physical-Layer Authentication (PLA), which focuses on achieving endogenous security and intelligent identity authentication, has attracted considerable interest. When compared with Discriminative AI (DAI), Generative AI (GAI) offers several advantages, such as fingerprint augmentation, reconstruction, and denoising. Inspired by these innovations, this paper provides a systematic exploration of GAI's integration with PLA. We commence with a concise review of identity authentication techniques and GAI models. Then, we contrast the limitations of DAI with the potential of GAI in addressing PLA challenges. Specifically, we introduce a structured taxonomy for GAI-enhanced PLA methodologies, encompassing three key stages: fingerprint collection, model training, and performance optimization within the PLA pipeline. Furthermore, we propose a novel PLA framework based on GAI and channel extrapolation for dynamic environments. To demonstrate GAI's efficacy in enhancing PLA robustness, we implement a case study using the Generative Diffusion Model (GDM). Finally, we outline potential future research directions for GAI-based PLA.