"Login failed for user" when using admin user in SSMS
Pretty sure the problem is in the username. When you connect to an Azure SQL Database instance, foo@bar implies that you're connecting as the login "foo" to the instance "bar.database.windows.net".

The credentials you use when logging into the Azure management portal won't work when you're using SSMS.

Verify that you've set up a SQL Database login, and use that login. The admin account is set up in the Azure portal - you can verify the login name by clicking "Properties" on the server instance.

Regular logins/users are set up as 1) logins on the database server, or 2) as contained users in each database (if you have enabled containment), or 3) in Active Directory (if you've configured one).