Compare: "We are working on Project Paperclip" and "We are working on a paperclip". I suppose the former implies that what you're working on is not a literal paperclip (but a secret operation to snatch scientists).

So "Project Gemini" is not about, say, he constellation.

With modern two-random-word codenames we tended to just say things like "I'm working on Crystal Banana all next week"

(Crystal Banana was a local joke codename where I worked)