You are expecting the zener current-vs.-voltage curve to be a step function: zero current below 2.0V, infinite current (as much as the supply can give to keep it exactly at 2.0V) above 2.0V.
In reality, unfortunately, the zeners are far from step functions. It's more like an exponential function. Look at the datasheet, the curve is printed there!
For that matter, any proper textbook should really explain this.
Additionally, what the "typical" datasheet curve doesn't show you, is:
1) Temperature variation: the voltage point where it conducts a certain current x is different at different temperatures!
2) unit-to-unit variation: similary, the voltage point where it conducts a certtain current, is different when you take the next diode of the batch; and even more different when you order a next batch.
For this reason, simple zener regulator is mostly useless for solar panel shunting, to prevent overcharging a battery, for example.
With a lot of margin and carefully selected, it may work but it would still end up discharging the battery each time there is too much shade.
The only way around this, unfortunately, is a bit more complex active circuit; for example a voltage reference IC like TL431 which can actually sink quite a lot of current (100mA).