Ah, this has been a time suck and frustrating overall.
Without a clear understanding of the issue, some users posted frustratingly online. This was obviously a new bug, as there was little online about the error this time yesterday. Since then, however, there have been other reports of the issue (for once, it’s not just me!).
You will know you’re getting this error if you go to authenticate to your google account and you are prompted to open Safari to authenticate… and Safari goes nowhere. A blank screen on Safari is opened and then it quits.
As Tasha (Litt) Jameson describes: “Google requires completing authentication in Safari. After authentication, set up will continue in Internet Accounts.” When I click “Open Safari” a blank page loads in Safari, and that’s it.
What we know is, it appears that updating to Mac OS Mojave 10.14.6 enabled an update of Safari to version 13 — aaaaand Safari version 13 seems to break the authentication loop for Google accounts with Two Factor Authentication.
This bug has been filed with both Google and Apple, and a temporary workaround has been posted here involving using IMAP for email and notes access for the moment, though this leaves iCal users (like me) up in the night.
For the moment, since I need calendar to run my business normally, I will try this advice to reinstall Mac Mojave 10.14.6 and restrict the install of Safari 13 for now (and I use Chrome anyway). I’ll keep you posted.