Can I use two LED in series without a resistor?
Since 2XLED voltage is ~5v, can I use it without resistor(i tested
  this and LEDs did light up).


Many LEDs have a negative temperature coefficient of forward volt drop versus temperature and, if the device gets warmer, it may disproportionately take more current and get warmer and warmer until the LED fails. You must satisfy yourself that this does not happen and I can't find any info in the data sheet to state what the tempco is so it's a bit tricky to guess. I would use as much resistance as I could afford.


  I checked forward voltage using diode voltage function in multimeter,
  that shows a voltage of 2.3 V, why is it different from the datasheet?
  Is it common?.


Look at the graph for typical forward voltage: -



At 10 mA the forward voltage (typical) is 3.0 volts but, your multimeter will probably use something like 1.0 mA as a diode test current so what will be the typical forward volt drop then? I estimate from the graph it will be 2.625 volts or about 88% of the volt drop at 10 mA.

Now, if the device you chose had a typical forward voltage of (say) 2.7 volts at 10 mA then it might only register as 2.36 volts on your meter.

So, the value you measure on your meter depends on several things.


  If putting it in series without resistor is not a good idea, can you
  please suggest a recommended method?


Plenty of folk use constant current feed circuits.