This issue is raised and addressed ad nauseam on HN, but here goes:
It doesn't mean anything when a model tells you it is ChatGPT or Claude or Mickey Mouse. The model doesn't actually "know" anything about its identity. And the fact that most models default to saying ChatGPT is not evidence that they are distilled from ChatGPT: it's evidence that there are a lot of ChatGPT chat logs floating around on the web, which have ended up in pre-training datasets.
In this case, especially, distillation from o1 isn't possible because "Open"AI somewhat laughably hides the model's reasoning trace (even though you pay for it).
So how is it then that none of the other models behave in this way? Why is it just Deepseek?
It's not distillation from o1 for the reasons that you have cited, but it's also no secret that ChatGPT (and Claude) are used to generate a lot of synthetic data to train other models, so it's reasonable to take this as evidence for the same wrt DeepSeek.
Of course it's also silly to assume that just because they did it that way, they don't have the know-how to do it from scratch if need be. But why would you do it from scratch when there is a readily available shortcut? Their goal is to get the best bang for the buck right now, not appease nerds on HN.