Generative Artificial Intelligence (GenAI) has revolutionized content creation across multiple domains, enabling novel applications with emergent reasoning and problem-solving capabilities. However, this rapid advancement raises critical privacy concerns from a user perspective, especially as GenAI models rely on massive data collections. This paper examines the concept of user-aligned privacy in GenAI, emphasizing the importance of capturing user preferences, ethical considerations, and the challenges of transparency and control. To further explore these issues, a user-aligned survey is conducted to explore individual users perception of privacy in GenAI and outline directions for future research. Furthermore, this paper proposes a framework to empower users with privacy management tools tailored to their needs and discusses regulatory and ethical implications, including recent legal and regulatory frameworks such as the EU Artificial Intelligence Act.