Could it be because Claude / ChatGPT are the most viral models and thus are more likely to be in the training set so when someone asks "Which AI are you?" the likeliness of Claude and ChatGPT being a reasonable answer is higher?
It has been observed again and again that ChatGPT and Claude would identify themselves as DeepSeek when asked in Chinese, through the API without a system prompt to solidify their identities. I don't think identity is a good evidence for distillation.
An alernative, it could also be the case that all of the model developers are using distillation to improve their models while crying do as I say, and not as I do.
Or the answer is even easier. There is public data consisting of chat transcripts with ChatGPT, Claude and DeepSeek, and all AI providers train on that data. You don't need distillation for the text "I am ChaGPT/Claude/DeepSeek" to often appear in your trraining data.