I have created a second database on my existing Azure SQL server. The first database works fine and I can see it using SSMS.
I cannot see the second database in the object explorer. Autocomplete detects that it exists however.
So for me, it turned out that when I was connecting using SSMS, I had set the database to connect to as my first database by accident - meaning that was all I could see.
On connect, go to Options, then check the database you're connecting to. If you're using multiple users to connect to that server like I was, my admin user ended up also being forced to connect to a single database instead of < default >.
SSMS does not automatically update the database list when new databases are added. You need to refresh the view manually.
In Object Explorer, highlight the Databases node. Hit F5 to refresh. The list of databases should refresh and your new database should appear.
Another method would be to disconnect SSMS from the server and reconnect.