Self-anthropomorphism in robots manifests itself through their display of human-like characteristics in dialogue, such as expressing preferences and emotions. Our study systematically analyzes self-anthropomorphic expression within various dialogue datasets, outlining the contrasts between self-anthropomorphic and non-self-anthropomorphic responses in dialogue systems. We show significant differences in these two types of responses and propose transitioning from one type to the other. We also introduce Pix2Persona, a novel dataset aimed at developing ethical and engaging AI systems in various embodiments. This dataset preserves the original dialogues from existing corpora and enhances them with paired responses: self-anthropomorphic and non-self-anthropomorphic for each original bot response. Our work not only uncovers a new category of bot responses that were previously under-explored but also lays the groundwork for future studies about dynamically adjusting self-anthropomorphism levels in AI systems to align with ethical standards and user expectations.