This seems to be something new, or at least since the last time I created an ODBC connection to SQL Server from Minitab 16.
When I created a new ODBC connection, the database name is now in the dropdown list.
As you can see, I can't read the full table name, and several of my tables have similar names. It didn't always include the database name, is there some option I need to uncheck or something to remove it?
It turns out this is a function of Minitab and there's no way to actually change it. It's just how the program was designed.