I have exactly the same issue described in this thread:
I have a Nest thermometer. I migrated from username and password to login with my Google Account.
I want to register my wife as a user, and Nest informed me I must use the Google Home app. So I installed this and logged in with the same Google account. It does not show any existing devices
When I select Nest and add it, it asks me to log in to my Nest account with a username and password. I use the username and password of my Google account which is rejected. It then prompts me in red to say that I should sign in with Google.
However, there is no "sign in with Google" option on the page.
Any ideas how I can either get my Google Home app to recognise that I have a Nest thermometer, or to sign in and link it?
This page addresses issues with migrating from Nest to Google: https://support.google.com/googlenest/answer/9297676?hl=en
When you install a Nest Learning Thermostat or Nest Thermostat E in the Google Nest app, it is supposed to AUTOMATICALLY appear in the Google Home app. However, in recent months, I've seen many customers in this forum report this isn't happening.
Many of these customers have done the following to fix the problem: In the Google Nest app, create a temporary second Google Nest "home/structure". In the Google Home app, go to that empty second Google Nest "home/structure" and delete it. This seems to force the 2 apps to sync and the thermostat then appears in the Google Home app.