If you want to define volume in D-dimensional space then you need to choose what the unit volume is, for example that a unit hyper cube has unit volume one. (That choice is relevant for these discussions and you could equally have chosen a unit ball to have volume one since you're defining D-dimensional volume you get to choose.) But a probability distribution is always normalized to integrate to one, which means either choice of volume definition will give you the same uniform probability distribution on the ball. So there's an extra arbitrary constant to choose for volume that isn't important in the probability distribution even though you define the probability density in terms of the volume.
