@gg51
If you have 1st gen Google Nest cameras and doorbells and have been using the Google Nest app and are logging on using the "Log on with Nest" option with a non-gmail account and have not yet migrated your Nest Account to a Google Account, then you should have first migrated your Nest Account to a Google Account (https://support.google.com/googlenest/answer/9297676) before installing your 2nd gen battery camera if you want all of your 1st and 2nd gen cameras and doorbells to be on the same account and in the same Google Nest "home/structure".
In your case, seem it SEEMS you have installed your new 2nd gen battery camera in a Google Nest "home/structure" in the Google Home app using one of your Google (gmail) Accounts (the one you refer to as "a random email account"). If that is the case, then you will probably need to follow the instructions in the "Error message: Google Account is already linked with another Nest account" subtopic in the link I provided in order to get all of your 1st gen Google Nest cameras and doorbells into the Google Nest "home/structure" on the Google Account where you installed your 2nd gen battery doorbell in the Google Home app.
If you want help from Support on this:
1. Go to https://support.google.com/googlenest/gethelp.
2. Select a category/product family, and type in a brief description of the issue, then click "Next".
3. Click "Other", then click "Next step".
4. Under "Resources", just click "Next step". (Clicking one of the listed links will bring up a Help page.)
5. Under "Contact options", choose to get a phone call or chat with the team, depending on availability in your country.
