The Database Key for the preview version of an App Maker app can be found in SETTINGS>DATABASE as the Database Key. In GCP, App Maker uses a SQL instance, and within that instance many Databases are listed - including the Database Key/Name from the App Maker preview.
By default, when publishing an App Maker app, a new Database within the MySQL instance is created.
How do I find the Database Key/Name that the newly published app is using?
You need to go to the deployments settings and click on the EDIT option. Then you should be able to retrieve the database key. Here, for your reference:
This is documented right here: https://developers.google.com/appmaker/deployment/#databases