He began the conversation by saying: “so as far as I know there’s two kinds of ai, open ai and closed ai…”
This speaks to your point, the people are confused.
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.
"Why is your French coworker helping you write literature? I thought you worked in software"
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
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).
If it has Open in the name it's something to do with open source and "AI" right? :)
Well if that's all that's at stake here, it seems very reasonable.