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On Firefox at least that flow says it's only for Android devices.

> Allow Google to display a sign-in prompt on Android. Learn more about sign-in prompts on Android[0]

with a section below saying Chrome itself has settings as well

> To manage third-party sign-ins on Chrome, go to Chrome Settings. Learn more about sign-in prompts on Chrome[1]

I followed the above link, and it seems that it's a Chrome setting now: https://support.google.com/chrome/answer/14264742

Will setting that other than default default give the site one more bit to fingerprint you?

Looks like that only works if you’re a Google customer?