I found the solution.
ASP.NET was not sending the cookie because the browser capabilities object for the java user agent Mozilla/5.0 (Java 1.6.0_17; Windows 7 6.1 amd64; en_US) ICEbrowser/v6_0_2 returns false in the Cookies field.
I created a new java.broser file and put it in the App_Browsers directory. I redeployed the application and it worked! ASP.Net sends the cookie to Matlab now.
What I had to do next is to modify the Matlab Soap function so it would receive and resend the cookie everytime. I posted the solution to that in http://www.cadec-online.com/Help/API.aspx#Matlab
I will report this issue to Matlab now.