Every AuraDB Instance is instantiated with a DB ID which is a unique identifier. There are a few ways to get this DB ID.
If you are using the Desktop client you will see the DB ID when you connect to the AuraDB Instance and select Open and then Neo4j Browser.
If you are using the web browser-based client you'll see the DB ID on the Instances page.
You can also check your logs and look for the connection URI which will be in the form of neo4j+s://<DB_ID>.databases.neo4j.io.