Why did Genghis Khan and the mongols cut one ear of every dead body?

Severing parts of a body was common practice in ancient times to prove the number of enemies killed, but there was never a consensus on the severed part. Duplicates are not much of a problem because left and right ear are different, even if the distinction requires some inspection. The tongue is not an easy organ to extract, and was, as far as I know, mainly used to punish post-morten orators and satirists (e.g. Cicero).

But other considerations might take place in extracting an organ. After all, duplicates are not the only problem, your soldiers may cut their own dead friends members to inflate the death tool. An interesting case is the Egyptian army, who would usually sever hands to count. There is at least one instance in which, during a certain period, they would sever the enemies penises to count the dead tool. The reason was both cultural and practical. You inflict a final humiliation on the enemy and you make sure that soldiers are not cutting Egyptians penises to inflate the death-tool. Since Egyptians were circumcised, the distinction was very clear. Further connection to the Osiris myth can be made, but this might be a step too far.

There is nothing, to my knowledge, that would point to a special consideration for ears in the Mongolian culture, so the best answer I can formulate is in terms of practical considerations: it is easier and faster to cut 10.000 ears than 10.000 tongues or hands. Mongols were very systematic when executing captives, it would not be a stretch to imagine that they would take efficiency seriously when severing body parts to count the score.