Don't try to use dynamically named variables. That's simply not how variables work in C#.
Use an array of lists:
List[] user = new List[infoForUserSessions.Count];
for (int i = 0; i < infoForUserSessions.Count; i++) {
user[i] = new List();
}
If the number of sessions can change, you would use a List>
instead, so that you can add lists to it when you add items to the infoForUserSessions
list.