Exactly. One of my coworkers prefers Gemini to overcome the blank page hurdle, and he happily describe it as "the ChatGPT from Google". What does that mean for ChatGPT as a business? Nothing. Google would like people to use Gemini, but at least they retain this user and can target him with better ads, their real business. ChatGPT is just a layman synonym for LLM.
It amuses me that ChatGPT actually seems like a generic term already. You Chat with a Generative Pre-trained Transformer. Does what it says on the tin!
The USPTO agrees with you.
"[Trademark] Registration is refused because the applied-for mark merely describes a feature, function, or characteristic of applicant’s goods and services."
https://tsdr.uspto.gov/documentviewer?caseId=sn97733261&docI...