"Gemini" is a very common name (being the name of a constellation) which has been used by countless products, companies and endeavors over the years. Almost no one outside of Hacker News and a small core of misanthropic anarchists knows about, much less cares about, the Gemini protocol.
In the case of this specific Gemini, it's apparently the result of there being two teams involved, and it's a reference to the Gemini space program[0].
Interestingly, were it trademarked you wouldn't have this snarky attitude about how a name doesn't matter unless people care about it. I'm not talking on other forums right now either, so who cares about outside of HN while talking on HN?
¯\_(ツ)_/¯ You asked and I answered, and provided a reference for the answer, which was outside of HN.
And as far as trademarks go, it still wouldn't matter. As I mentioned, countless entities were already using "Gemini" before the protocol was created, and Google's AI isn't even in the same domain.
In the case of this specific Gemini, it's apparently the result of there being two teams involved, and it's a reference to the Gemini space program[0].
[0]https://twitter.com/JeffDean/status/1733580264859926941