I've only seen thankee on Everything2, so I do believe it is an Everything2ism. If it isn't, I still choose to believe it just like I believe in Santa Claus and the moon landing.
Thankee is a pseudo-shorthand way of saying thank you. Most of the time it is used in the catbox (the chatbox for newer folks and visitors) in a private message when one is thanking someone, typically for a ching or C!.
Some folks, like our dear departed dannye (rest in peace), preferred not to get thanked when he doled out his chings. Getting a Thankee seemed to slip by better because it wasn't a formal sign of appreciation.
Iron Noder 2017