I think it might have something to do with how much freedom a video gives you.
When reading a book or watching a movie, you have no input. The story follows the same regardless of what you want and do, so if you have some other ideas for it, you have to write your own version.
However, while playing a video game, you are in control. Not over everything, but if you want to abandon your world-saving quest to roleplay being a villager, you often can.
If you want to experience the video game world in a new way, you just need to change your game style. There are restrictions, but when you are playing a video game you are already writing your own personal story. Therefore, you don't get the same impulse to write fanfic.
This does not mean that there should be no video game fanfic, but it might explain why they are not as common as movie and novel fanfic.