Database Security

Yeah, this is the price you pay for cut rate SQL hosting. It's not the end of the world, but you get what you pay for.

The idea of Security Through Obscurity is usually considered a bad practice when relied upon on it's own. Rather it's a better idea to secure systems as if an attacker had full visibility of how things are setup. It's also important to note that the databases running on a GoDaddy instance are not visible to the public - they are visible to other paying users who have databases running on GoDaddy's servers. Not a high bar for entry but also not the public. That said - the security mechanisms that SQL Server makes available should be capable of isolating database instances from one another and to my knowledge there has never been a GoDaddy security breach related to the ability to see other database instances running on the same SQL Server instance.

There are a number of reasons to not use GoDaddy for database hosting, but I don't think that this particular oddity is one of them. Personally - I have used them for DB hosting in the past and I would not recommend it for anything but hobby projects.