When you have a 12 inch ruler, you effectively have 13 numbers on the ruler. The fact that zero isn't marked is neither here nor there -- the numeral one is not at the far end of the ruler.
So if you extend the ruler to be as long as you can hold in eight bits, it will range from 0 to 255, and the total length will be 255.
The ruler analogy may seem overly simplistic, but then the real world is likewise fairly simplistic.
At the end of the day, the numbers presumably come from a sensor, or go to a display, and, often, in either case, zero represents as dark as you can get and 255 represents as light as you can get, so the physics dictate that the intervals associated with the 0 and 255 are half the size of the rest of the intervals.
Audio is more interesting than video, because in audio, you care deeply about not having an offset, and about having a balanced signal, so the question of whether the midpoint is actually on a number or not is pertinent.
In audio, it is often useful to simply discard a code so that 0 is the midpoint (e.g -65535 to +65535, discarding 0xFFFF). But this still gives you smaller intervals at both ends.