No. On login Google shows you your email address so you can click on it and only fill the password. Which means it knows you were logged in on that browser previously.

That shouldn't be possible in private browsing, unless you've previously logged into Google in that same private session.

One way you could see something similar to this would be if you opened a clean session, logged into Google, logged out of Google, thought you closed the last incognito window but didn't, and then opened a new incognito window? Then the user cookie would still be in client-side storage

It knows that one from cookies, not applicable to incognito mode. Google probably does some other fingerprinting as well though.