The listener is really a way for clients to find the instance where the databases in the availability group are currently available. The listener itself is not an instance of SQL Server at all which I think is what you are thinking.

If you connect to the listener (e.g. with SSMS) and list the databases that are on the instance, then failover to the other instance in the availability group, re-connect to the listener and list the databases again, you should see a different list because each of the instances (probably) has different database on it.