I think you are confusing the particular implementation of the Singleton pattern with the concept itself. Your previous question / implementation could be interpreted as Singleton within a user session, if you saved in the Application cache instead it would be a Singleton for the ASP.NET application.
Singleton just means that there is only one instance of a particular object at any time and this is enforced by the application - how you enforce this is an implementation detail that might vary depending on your choice of platform / language.
In particular for ASP.NET you may still use the standard singleton pattern for C# using a static variable (see Implementing the Singleton Pattern in C#). There will only be multiple instances if your application is used in a web farm scenario, also refer to this SO thread: Are static class instances unique to a request or a server in ASP.NET?