Going to https://gemini.google.com works fine for me when not logged in. It might be doing some sort of reputation check on your browser/IP to decide whether it requires a login or not.
edit: sure enough, while using Tor or a well known VPN IP, Gemini requires I login.
That's not inconsistent with what I reported. It seems to require it sometimes, but not others, for mysterious reasons.
Are you and your colleague both trying at work? Probably on the same IP? Google might attribute less trust to an IP shared between many different users than it does to a regular residential internet IP (like mine).
Did some more testing and the behavior is interesting. When connecting through a Mullvad node in the US it doesn't require login, but any Mullvad node outside of the US and it does. I might be wrong its and it's just a per-country policy.
As for AI mode from google search I am not sure but I don't seem to have it, at least in my country, switzerland.