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).
or, hear me out, off and then…