If all the classes do exactly the same thing except they point to a different database, then the way you propose seems to be a whole lot of code duplication across several files for no reason (this will become a maintenance and testing nightmare).
I'd create a single class/module, and select the database when loading the module (or on object creation) based on whatever condition you need (environment variable, object instantiation parameter etc). For this, you can use if statements, a dispatch table etc.
Using if statements or a dispatch table is by far more elegant than having multiple modules all doing the same thing. Besides, even in your example you posted, you're still going to have to use if statements; at the module loading level instead of database selection level.