For Connecting to multiple database using single connection, the user name used in the connection string should have access to the database.

eg: Say we have two db Test1 and Test2 and the initial database in the connection string is Test1. For Selecting data from Test2 db, one should use the following query:-

Select * from Test2.dbo.Emp.

Again, this query will only get execute only if user has access to Test2 db.