I'm not sure what you mean. If this involves "sign in to Chromium" then that seems to involve signing in to Google, and that sort of defeats the purpose.
Incidentally, Chromium is what I use for Facebook...and only Facebook, on this particular computer. And signing into Chromium with my Google account seems to me to be a big information leak.
Chrome/chromium has the distinction of user profiles[1]. You can have separate windows have separate profiles (but not separate tabs). This has nothing to do with Google accounts. Each profile has their own 'jail' of chrome resources (cookies/history/tabs/etc). This is really useful when you need multiple persisting sessions while doing web dev or whatever else.
Incidentally, Chromium is what I use for Facebook...and only Facebook, on this particular computer. And signing into Chromium with my Google account seems to me to be a big information leak.