If you look at the datasheet they will generally give a typical voltage at a given current, most multi-meters use a constant current source, something between 10mA and 100uA to drive the diode, however these are not intended to tell you everything about the diode, more that the part is working as intended,
Your main deviation would be the differences in test current, to find this out, put one meter on diode test, and another on mA, and cross connect the 2,