Recent studies have shown that the outputs from large language models (LLMs) can often reveal the identity of their source model. While this is a natural consequence of LLMs modeling the distribution of their training data, such identifiable traces may also reflect unintended characteristics with potential implications for fairness and misuse. In this work, we go one step further and show that even when LLMs are trained on exactly the same dataset, their outputs remain distinguishable, suggesting that training dynamics alone can leave recognizable patterns. We refer to these unintended, distinctive characteristics as natural fingerprints. By systematically controlling training conditions, we show that the natural fingerprints can emerge from subtle differences in the training process, such as parameter sizes, optimization settings, and even random seeds. These results suggest that training dynamics can systematically shape model behavior, independent of data or architecture, and should be explicitly considered in future research on transparency, reliability, and interpretability.