All languages I know of have four basic directions: left, right, backwards and forwards. Are there any that don’t?
Maybe you could have just three: forwards, backwards-left and backwards-rigth? Or more then four, maybe with distinct names for some or all of the four in-between the common ones
The number of dimensions in space is three, and that has nothing at all to do with language. When you eliminate one of the dimensions (“up” in your example), you are left with two, and you’re describing dimensions in a plane.
The reason all languages use four words for the two dimensions is because there are two directions per dimension.
ETA: why there are three (plus one for time), and how do we know that, are very interesting questions (to me).