I mean, I get the rationale Company vs. Product, but most people know the product. As in "I used ChatGPT". But if you ask who OpenAI is, they'll have no clue.
ChatGPT is in someways nicer... because their models are GPT-5.3, GPT-5.4, etc...
But when you're trying to explain that the Anthropic models are called "Opus" or "Sonnet" or "Haiku" or "Fable", but you use them in "Claude", it gets confusing quickly.
As such "everyone knows them" isn't a reason to allow a registration. It would just mean that blocking the trademark has no practical effect
If it has Open in the name it's something to do with open source and "AI" right? :)
On a side note, the AI models from the company are not even open, one can go as far as banning it as inappropriate marketing (Product not matching the description).
Well if that's all that's at stake here, it seems very reasonable.