Are the values of Shared Properties outside of aspx pages Global to all sessions?
Properties are just get and set methods. It's static fields that are shared by the app domain. Since you have no fields (only wrapping a session variable in get and set methods) everything is scoped to the session.
Note that automatic properties (e.g. Shared Property MyString As String in VB) actually create backing fields that you don't see. But that's not what you're doing here, so no backing field is created and no static data is used.