The "diode" setting on your multimeter is really an "ohms" or resistance function except, instead of resistance, it displays the voltage across the two probes while providing a very small constant test-current from a low-compliance (typically less than 9 V, often less than 1 V) current source within the multimeter. For example, a typical ohms function test current might be one milliampere, causing a 1000 ohm resistor to drop one volt, which the meter would then interpret and display as 1000 ohms. Because the constant-current compliance voltage is so small, higher resistances must be measured at smaller constant currents. Thus a ten megohm resistor might be measured with a constant current of 0.1 microamperes, resulting in a drop of one volt across a 10 megohm resistor.
Now, switch to the diode function and what happens? First, the meter no longer displays resistance; it displays the voltage between the two probes
as a voltage. Second, the current supplied can be larger than the current used for resistance measurements, but is still limited by the internal battery to something on the order of one milliampere. It is definitely
less than the typical operating current of an LED. What is your take away from all this? The diode function on a multimeter is good for several things, but testing the
operating forward voltage drop of an LED is
NOT one of them. Use the diode test function as a Go/No-Go test to find shorted and open diodes and as a way to distinguish germanium diodes from silicon diodes. If you know the polarity of the test probes, you can also determine which terminals of the diode are cathode and anode.
The forward operating voltage of an LED depends on two things: how it is constructed and the forward operating current through it. You can possibly use the diode function to determine how the LED is constructed. Red LEDs will have a lower forward voltage (at any forward current) than blue, green, yellow, or so-called "white" LEDs. You can use that information in conjunction with a datasheet to "ballpark" a safe current-limiting resistor and power supply voltage.
No datasheet? Sneak up on the maximum LED operating current by using a variable constant-current power supply, as
@BobK suggested, or start with a high-valued current-limiting resistor and a variable voltage power supply. Work toward lower resistance values. Stop when you reach the brightness you need, or the LED burns out, whichever comes first. Measure the current through, and the voltage drop across, the LED where satisfactory operation occurs. Make a note of it for future reference because there are an infinite number of voltage supply and current-limiting resistor combinations that will work for that particular LED. You can use the information you just determined to calculate them.
It baffles me why so many people on this forum think powering up an LED is akin to rocket science. It's really quite simple, and
we have a nice resource here that describes how to do it. Strongly suggest you swing on over there and read it.