Web Apps in general do not keep track of separate machines accessing them, they just keep track of different browser Sessions. Each instance of a client connecting to your app will generate a new session, and the information about this session is generally managed via a special cookie named ASP.NET_SessionId. The configuration for tracking the session allows changing the cookie name or to use non-cookie tracking. This tracking of Sessions can be utilized in a few different ways to differentiate between the separate browsers.
One method would be for your application to keep track of the session count. This would be done via global.asax and adding to the different application and session events. There are a few answers in the SO thread titled How to count Sessions in ASP.Net.
protected void Application_Start() {
Application["LiveSessionsCount"] = 0;
}
protected void Session_Start() {
Application["LiveSessionsCount"] = ((int)Application["LiveSessionsCount"]) + 1;
}
protected void Session_End() {
Application["LiveSessionsCount"] = ((int) Application["LiveSessionsCount"]) - 1;
}
And when it is time to set your cookie; you could access that Application object to retrieve the count, and use that to create a different cookie value.
int LiveSessionsCount = (int) Application["LiveSessionsCount"];
string MyCookieValue = "V" + LiveSessionCount.ToString();
Response.Cookies["MyCookieKey"] = MyCookieValue;
The problem with this would be that the application is keeping track of current sessions. This will increase and decrease as connections to your site start and end. What you could do would be to add a second Application variable that has similar methods in the App/Session starts to start at 0 and increment; however, do not add in the Session End decrement of that value. This would then be a total count of sessions within the app cycle. When the site is restarted this would return to 0.
There are naturally other methods you can use, one of the sites I work with logs all connections into a database and worked with that way. This is just to give you some ideas on what can be done. I would read the documented links to get a better understanding of the application and session objects; as well as the session cookie.