Sounds like Google doesn't create per-account cookies and uses one cookie set per browser instance, so you're either logged in or you're logged out, one account at a time.

Browser containers or profiles might be the answer to what you need.

I've been using different Firefox profiles to run accounts in parallel for well over a decade at this point. Containers might be able to do the same thing, but on account of the aforementioned, I've stuck with the old way of doing things.