yeah well i had this discussion before in regards of Babur, because of a passage in the Baburnama that implies something we call "bicurious" today, and because he rarely talks about his wifes or love at all. i wouldn't read too much into those things, precisely because of our modern liberal and open-minded societies.
the examples you brought up are particular to those cultures, in this case the Greek and Japanese (Chinese also had that). it's wrong to conclude that any culture had that.
I totally agree with you, but then that should be also true for the "modern mongolian culture" which is inevitably different from "gingis khan's time mongolian culture" as much as is different the japanese or the greek culture (maybe even more different). Therefore it is also wrong to conclude that it is impossible that he had a male lover only because it sounds strange and incredible to our modern idea and our modern perception of homosexuality and of gengis khan himself.
The only correct approach is to observe other cultures, to understand that a warrior having a male lover is not so strange and unusual as it could appear to us, and then to approach the original sources of his time (the sources about him in particular, and about the culture of his time in general) with a mind open to both the possibilities.
There is a mention in "Jagfar Tarihï" that homosexuality was practiced amongst Mongolian soldiers, but I am not sure if this should be taken literally or as intended offence since the source itself was composed in Muslim milieu with strong anti-Mongolian attitude.
From the other hand in the Yasaq is stated that homosexuality is punishable by death.