Why can't I view the SQL Server database created by a web application?
Please stop using the user instance=true setting. Not only has this feature been deprecated, but what is actually happening here is that Management Studio is firing up a different user instance of SQL Server than the one your application is using, so it is not surprising that you can't see the same thing in both cases.

(Usually, the symptom of this is that someone runs an update from their app, then checks the table in Management Studio, thinking there is a bug but not realizing that the update affected a different copy of the database.)

Create the database on your running SQL Server instance (or attach it there). Then connect to that server the same way, without the User Instance setting, from both SSMS and your app. Lots more information here. Don't create the database every time the application starts - that just sounds horrendously wrong.