It does feel like there is something that happens to people when they ask an LLM to name itself. I don't think it's inherently bad, but it seems to be a common theme with people whose interactions with LLM border on (or cross into the) delusional.
Tangentially, I noticed recently that I'm always fairly respectful in my LLM prompts and often say "thank you" as part of them. LLMs don't need that, but I've come to realize that I'm saying those thing for me. I don't like being disrespectful and expressing gratitude is important to me. Is expressing gratitude to a LLM strange? Perhaps. Is it harmful though? I don't think so.
But yeah, asking an LLM for their name? That seems like something else.
I recently saw a comic where the machines took over and "that guy" (you) in the comic was the one the machines allowed to live because "he was always polite". ;)