There are a few issues with how you think of an account associated with activities on Google and Google Home. Google Home does not give you direct access to your Google account — instead, it acts like a web browser, and the account build process requires you to issue the OAuth2 token in your “browser” at home so that it can use it in the future.
If you have control over the external API and it issues OAuth2 tokens (which looks like this), you can completely skip the Firebase part. You just need to configure the API.AI with OAuth2 information for this external service - client ID and secret, URL for the login page and token exchange page, etc. In this case, your web hosts will be called by providing OAUth2 an access token that you must pass to the external API when you call it. For more information, see the documentation for the "Actions for Google" Account documentation .
If you are not in control of this API, you may need to provide a basic OAuth2 server implementation that can issue authentication tokens (those that you create, or those that can be used to get auth tokens from an external API). Then your web sites will be called using these OAuth tokens, and you must use the token to find the token to access the external API. You have some options for implementing this, and these options are discussed in the OAuth2 Account Linking Overview in the Actions docs for Google.
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