Which Chrome API or other Google API is Pinterest exploiting here to detect visitor email addresses?
"Use pinterest.com with Google"
I believe this is Google Identities "One-tap sign-up" feature.
With Google's one tap sign-up and automatic sign-in APIs, new users can sign up to your site with just one tap on a dialog that's inline with your page's content. Returning users are signed in automatically, even when they switch devices or platforms, or after their session expires. - Google Identity case studies
It seems that this is currently a beta API feature which they are currently improving cross-browser functionality.