Yes, of course. Just because you have three dimensions doesn't mean you have to use them all.
You can use an isometric projection (which is a setting on the camera) to remove perspective and essentially ensure that things are the same size on screen regardless of how close to/far away from the camera they are.
If your game needs complex physics, you can either use a 2D physics engine or use a 3D physics engine and "manipulate" it to restrict it to 2 dimensions. For a platformer like NSMB you probably wouldn't need a physics engine and could write the game logic yourself, in which case it only uses the axes you choose to use.
These days, the majority of games are 3D (from a technical standpoint) regardless of whether they're actually playable in 2 or 3 dimensions. Even pure 2D games with no models or anything are probably being rendered in 3D, so choosing to add in 3D models or whatever is no problem at all. In certain cases, I imagine this would be very useful. If you have characters with a lot of animation, 3D animation and rendering is much more efficient and potentially much less expensive.
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