For those interested in source code, one can review Chromium's implementation of this feature in chrome_signin_helper [1], dice_response_handler [2], and adjacent source files in the /chrome/browser/signin/ folder [3], as well as the files in the /components/signin/core/browser/ folder [4]. To my eyes, it seems an API call is made from the browser to Google to obtain the signed-in state.

[1] https://chromium.googlesource.com/chromium/src.git/+/master/... [2] https://chromium.googlesource.com/chromium/src.git/+/master/... [3] https://chromium.googlesource.com/chromium/src.git/+/master/... [4] https://chromium.googlesource.com/chromium/src.git/+/master/...