One reason (but probably not main) is that if it wasn't this way it would not always agree with the euclidean conception of distance. Imagine you are asked for the points within a certain distance from one given point and such that the three of them are on a straight line. Intuitively, there are two solutions (each one at "both sides" of the given point). If the norm was the inner product raised to one, then the answer would be a singular point and while this might look OK in an abstract sense, it would not agree with "reality", if you want to call it that way. I mean, you would be missing half the solution to the problem.