File this in the category of error messages that don’t really mean what you think. When configuring a failover group using Azure SQL DB, you need to provide a “name” for the failover group. However, the name really becomes the FQDN of the failover group, but that’s not directly obvious. The implication of that is that the name of the failover group must be unique, globally.
So when you see this error message, it’s not because you already have a failover group with that name – it may be because anyone has created a SQL Server or a failover group by that name.
I have filed a Connect request for Microsoft to add “.database.windows.net” below the text box like what is present when creating a new server (below).
If you feel the same way about this as I do, you can vote for it here: https://connect.microsoft.com/SQLServer/feedback/details/3137859