Copyright protection for large language models is of critical importance, given their substantial development costs, proprietary value, and potential for misuse. Existing surveys have predominantly focused on techniques for tracing LLM-generated content-namely, text watermarking-while a systematic exploration of methods for protecting the models themselves (i.e., model watermarking and model fingerprinting) remains absent. Moreover, the relationships and distinctions among text watermarking, model watermarking, and model fingerprinting have not been comprehensively clarified. This work presents a comprehensive survey of the current state of LLM copyright protection technologies, with a focus on model fingerprinting, covering the following aspects: (1) clarifying the conceptual connection from text watermarking to model watermarking and fingerprinting, and adopting a unified terminology that incorporates model watermarking into the broader fingerprinting framework; (2) providing an overview and comparison of diverse text watermarking techniques, highlighting cases where such methods can function as model fingerprinting; (3) systematically categorizing and comparing existing model fingerprinting approaches for LLM copyright protection; (4) presenting, for the first time, techniques for fingerprint transfer and fingerprint removal; (5) summarizing evaluation metrics for model fingerprints, including effectiveness, harmlessness, robustness, stealthiness, and reliability; and (6) discussing open challenges and future research directions. This survey aims to offer researchers a thorough understanding of both text watermarking and model fingerprinting technologies in the era of LLMs, thereby fostering further advances in protecting their intellectual property.