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The training data likely references Claude significantly more often than Kimi, given the popularity of the models. There will simply be more examples of “Claude” being the response to that question.

Doesn't Claude say its Deepseek when asked in Chinese? I remember there being posts about that a while ago.

Unless it is specifically instructed in the system prompt it will give you the most likely answer, which is Claude. If there are instructions in the system prompt it will give the correct answer, which would be Kimi. Or are you suggesting that Kimi is copy pasting Claudes system prompt?

Okay, but I would also ask “why does Claude say that its name is Claude?”

Because it's in the system prompt

AFAIK, you cannot run Claude without one of their mandatory system prompts at all.

If we wanted to compare model responses, we would give all models system prompts with model names, thereby fixing the Kimi misattribution.

The reason Kimi often states its name as Claude is likely because we can actually run it without the mandatory system prompt, smoothing over awkward competitor mentions.