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.