Large Language Models (LLMs) show impressive conversational abilities but sometimes show identity drift problems, where their interaction patterns or styles change over time. As the problem has not been thoroughly examined yet, this study examines identity consistency across nine LLMs. Specifically, we (1) investigate whether LLMs could maintain consistent patterns (or identity) and (2) analyze the effect of the model family, parameter sizes, and provided persona types. Our experiments involve multi-turn conversations on personal themes, analyzed in qualitative and quantitative ways. Experimental results indicate three findings. (1) Larger models experience greater identity drift. (2) Model differences exist, but their effect is not stronger than parameter sizes. (3) Assigning a persona may not help to maintain identity. We hope these three findings can help to improve persona stability in AI-driven dialogue systems, particularly in long-term conversations.