Well, the videos are using the 'fixed camera' mode. The Battle arena is 3D, of course.
About that 2D effect, I plan on releasing 2 sizes, and 2 camera modes per size. I mean, a natural camera, and a 'fixed camera' for the normal brawl, so you don't need to enter the 'fixed camera mode' to have the non-moving stage just like in the DK64 multiplayer.
EDIT: Well, was talking about the stage I've made, but I guess you meant the original multiplayer stage. It's 3D too. In some videos in youtube, you can see some holes in the structure, and those tell you it's 3D.
This video is an example: