Holy thread revival, Batman! (sorry, this was the most recent one describing this issue I could find).
I first encountered this problem about 6 months ago, gave it up as a bad job and at some point accidentally fixed the problem. I recently had to reinstall Kodi and encountered the same issue and after about an hour of wildly stabbing in the dark found that if you log in to a non-google site or service using the "Sign in with Google" option (for instance Duolingo), Chrome then automatically signs in.