You can change the name of a database which users see in the interface. Renaming the database changes the display name, not the names of the SQL files. Once you rename a database, you can check which SQL database it is connected to by checking the connection strings netpub/wwwroot/getsmart/web.config file on the SQL server. The Initial Catalog entry will tell you the database Name in SQL, while the "name" is what it is called in GlobalSearch.
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