What I mean by “half signed in“ is that:
When you normally sign-in to your account, the account’s profile image will appear on the top right of the page when using a Google service. Clicking it will expand and show account-related options.
When I sign into my account through a private window, some accounts also seem to do this “half-sign-in“ thing, where the profile image is in the top right of the normal session. If I go into the account activity, I can see that it’s tracking search history in the non-private window - the one that’s not supposed to be signed in.
This is an image of what’s shown on the normal window, after only signing-in to the private browser. Clicking any of these options prompts you to sign-in though
I would say that it’s just Google being Google, but I’ve tested multiple accounts with multiple browsers/versions, and nothing is consistent