Gmail is “free” because you are tracked by Google for their data/marketing purposes and are automatically signed into everything else as a result.
Hate to break the news to you but when you sign into Google it is not for one specific service; that is just a “carrot on a stick” to get you to login. The beginning, middle and end goal of Google’s “free” services is to just get you on their system to begin with.
Meaning there is no way to turn any of this off via a client side setting: If Google wants you logged into all Google “apps” and not just one of them—like Gmail—that is the way it is.