I tried opening Chromium on Linux yesterday (I use it sometimes for testing) and it prompted me to log in. I had to hit Cancel four or five times before being able to browse. Bizarre behaviour for a browser.
I haven't figured out a way to do so, at least. You can use guest tabs, but then you lose all of your customizations (extensions, bookmarks, etc.), so I don't consider that a viable option.
ungoogled-chromium removes this, but the current builds fail on my machine for some reason.