You have a separate database for each customer? That's an unusual way of doing things. Normally, you would using a unique ID for each customer and use the ID as a key in tables of customer data. Having customer data in a database for each customer makes it so that you can't, for example, run any queries that return information about more than one customer at a time.
But ... if having a separate database for each customer is what you really want, you can't get around selecting the database each time.