Why doesn't my multi-meter diode tester work on this LED?
Those LEDs have a typical forward voltage of 3V.
Your multimeter's Diode/Continuity test feature reads anything above 2.8V as an open circuit.

from this link

The Diode Test function on multimeters isn't really intended to test LEDs. Normal diodes have a forward voltage under 1V.  

'Old' Red/Yellow/Green LEDs would typically have forward voltages in the 1.2V to 2.5V range, but these new-fangled white/blue/purple/pink thingies are often 3V or more.