Why does a Zener diode leak so much current?

On a log scale, 5 uA is NOT EXCESSIVE and is consistent with the exponential behavior of all current rectifier diodes with a straight line except near the knee current where the bulk resistance dominates the curve as a linear slope.
The value of the linear slope is what defines the value of Rs and tolerance of Vf above 1 mA . This is the incremental resistance (from bulk resistance) or slope in V/I. "onsemi" calls this Rs , the Zzt value or rated threshold resistance. (Zzk is the lower current knee threshold) The rated current, Rs = Zzt are inversely proportioned to the size and power rating of the diode Rs=<=1/Pmax and for this part ranges in the 50-ohm range so its Vf rises significantly above 5mA as such. 0.75V/15mA = 50 Ohms (results from 20-5=15mA)
The linear plot below on the right Y-axis shows the Z value 1 to 5 mA and drops down from 5 to 20 mA.

As a matter of comparison a 5 mm White LED has a threshold voltage of 2.85V with a nominal Vf of 3.1V @ 20 mA with a supplier depending tolerance, thus the Rs or Zzt or bulk resistance is around 12 Ohms much lower than this Zener but makes a cheap regulator and indicator and is more consistent between suppliers if kept in the 2 to 3 mA range at 2.9V. Poor suppliers in the past have rated their 5mm LEDs up to 3.6V at 20 mA. The chip itself is much smaller than 5mm, which is just the lens.
This is an XY question from the comments and lack of clarity in the query text.
- the above correctly answers the question in the title.
- The perceived question is how to improve a 4-20mA to voltage converter with adequate protection for uC input, such as over-voltage or over-current with Vdd= 3.3V +/- TBD tolerance.
- Since the ADC has a high Zin and the load is low 160 ohms, all you need is a >=10 k series R from the 160 and then optional Schottky diode clamps to Vdd,Vss or rely on ESD diode pairs internally. If there is still common mode noise, then a differential Op-Amp performs better than a single-ended input to local ground. This often occurs due to SMPS CM noise with floating supplies.