MVC4 Multiple Databases

As I understand your question, the core issue is you are struggling to connect to a different db, whenever user selects a different product from a DropDownList. As you said, yes DbContext object requires a connection string in the Web.config file. You can specify multiple connection strings in a config.
Also you can definitely pass different connection strings to DBContext constructor. Typically your DAL/Data Access Layer or Repository layer would pull appropriate connection string from the Web.Config/App.config and pass it to the DBContext constructor. See a similar approach here and here.

UPDATE :

You cannot share the same DbContext with multiple databases. You need multiple DbContexts for each of your DB.

Additional There are few of doing this, but if you use Repostory and Unit Of Work pattern, you can use an approach like this

Each DbContext you going to have, you can associate with set of entities within that database in context. Something like below

public class ProductContext : DbContext
{
    public ProductContext ()
        : base("connectionStringA")
    {
    }

    public DbSet<Product> Accounts { get; set; }
}


public class LicenceContext : DbContext
{
    public LicenceContext ()
        : base("connectionStringB")
    {
    }

    public DbSet<Licence> Licenses{ get; set; }
}