Most meters use a constant current of about 1mA for the diode test (the chosen current is usually in the range of 0.4mA to 2mA). The voltage across the diode or LED is then displayed.
LED datasheets use a current of 20mA to characterize the forward voltage drop.
Open circuit voltage varies from as low as 1.8V for some meters, up to 15V for a few others.
All of them can measure the drop out voltage of a normal diode (0.6V). Few are designed to light up a LED and measure its drop out voltage.
Connecting a current shunt in series in the test circuit to measure the constant current makes things even more difficult for the meter, increasing total drop out voltage.
As an example, testing a diode, gives a drop out voltage of 0.6V. With another meter introduced in the circuit to measure current, the displayed drop out voltage becomes 1.6V.
With a white LED, the drop out voltage is a lot higher (2.6V). Compliance voltage can be exceeded and the meter's constant current source cannot maintain the required 1mA.
I suggest you go and read about constant current sources.