The LED already has a built-in diode, so there's no need to add one. It's there to provide protection against reverse polarity, which can easily damage InGaN LEDs, especially the more exotic UV types, which are more delicate. A series resistor will suffice.
I don't know why it's a zener, rather than an ordinary diode, since the LED is bound to turn on first. Perhaps it's faster than the LED, so will absorb high current ESD, before it gets to the LED?