Maybe I missed something, but it's kinda obvious for me. The user is or will be logged in to the forum with his account, why wouldn't they, when the goal is to buy a Copper membership? So the forum knows which user it deals with. I don't quite believe that this forum uses browser fingerprinting tecniques. For what purpose specifically?
If you are constantly making use of Tor you may not actually notice that the forum saves some cache and other data because of the fact that every time you close and reopen the browser you get a new onion circuit and based on privacy settings you don't get anything saved eventually.
But browsers like Chrome save data and cache for stuff like sessions, and to improve load speeds. You can confirm that when you try to clear your history you can either keep those cache or not.