The PRO and Enterprise versions of Blue Spatial support connecting to multiple database connections. You can select a connection when creating a Map Service or Feature Service. The default database connection which was created on install can not be deleted since it stores metadata for Blue Spatial. The Add connection dialog has a button to connect to the database server. Once a connection to the server is established you can select the database from the dropdown list and save.
For Azure database connections use the following format for the server name: tcp:[AzureSqlServerName].database.windows.net,1433