Recent Large Language Model (LLM) based AI can exhibit recognizable and measurable personality traits during conversations to improve user experience. However, as human understandings of their personality traits can be affected by their interaction partners' traits, a potential risk is that AI traits may shape and bias users' self-concept of their own traits. To explore the possibility, we conducted a randomized behavioral experiment. Our results indicate that after conversations about personal topics with an LLM-based AI chatbot using GPT-4o default personality traits, users' self-concepts aligned with the AI's measured personality traits. The longer the conversation, the greater the alignment. This alignment led to increased homogeneity in self-concepts among users. We also observed that the degree of self-concept alignment was positively associated with users' conversation enjoyment. Our findings uncover how AI personality traits can shape users' self-concepts through human-AI conversation, highlighting both risks and opportunities. We provide important design implications for developing more responsible and ethical AI systems.