In fact, it does vary. The finite slope of the Iz-Vz characteristic is expressed in form of a dynamic (differential) resistance rz which assumes rather small values (some ohms).
More than that, this value (the slope) is not constant but varies with the current Iz.
"Under a high reverse-bias voltage, the p-n junction's depletion region expands, leading to a high strength electric field across the junction. A sufficiently strong electric field enables tunneling of electrons from the valence to the conduction band of a semiconductor leading to a large number of free charge carriers. This sudden generation of carriers rapidly increases the reverse current and gives rise to the high slope conductance of the Zener diode."
So, a reverse biased diode has a non-destructive breakdown voltage, above which it has significant conductance compared to below it.