Why doesn't my multimeter show the correct Zener reverse voltage?
Because the limited current sourced by the meter is not enough to bias the Zener diode into the reverse avalanche region. Instead you are biased some where on the knee of the Zener reverse characteristic curve.

It is typical for meters to source about 1 mA in the diode test mode. Most Zener diodes will not reach their steep breakdown region until 10&nbsp;mA or more.

For testing Zener diodes in reverse bias you are much better off biasing the Zener diode via a resistor from a separate power supply and just using your meter to measure the voltage drop across the diode.

Be aware that meters in diode test mode typically limit the maximum voltage across the leads. This may be limited to the meter battery voltage or some lower value for better quality meters. This places a distinct limit on how much voltage the meter would show for reverse biased Zener diodes. It would be extremely uncommon to find a meter that could reverse bias a 27&nbsp;V or 48&nbsp;V Zener diode when the meter runs off a nine volt battery.