What do you think the Asian world would have looked like today if Kuhblai Khan had actually conquered Japan during the first or second invasion? Would history be significantly different without Japan being a distinct Asian entity?

Well, different to some degree but, the way I see it, the froces of Kublai Khan were rather stretched when they did try to invade Japan - they would doubtless have caused a lot of misery but, I doubt that it was possiblef or them to conquer the islands, their logistics were not all that good. Remember, the empire of Kublai Khan did not last all that long on the mainland either.
It takes some rather robust contrafactuals to construct a situation such that the troops of the Khan would have been able to handle the needed transports - they were not very good at shipping if I understand things correctly. Tried to ship the troops and supplies over the sea using boats normally better suited for rivers.

If you look at what happened between Poland, Moscovy, Novogorod and the Golden Horde, and Persia and the Timurids, not very much changed in terms of culture. They did however have a larger impact on China, finnishing of the Song Fynasy, which brought about the Ming Dynasty and indirectly Manchu. However, I don't think the Mongols where ever strong enough to defeat Japan in anycase.

My point exactly. So, for them to do so, many things need to be otherwise than they really were.

From what I understand, the Khan's armies had great early successes against the Japanese Samurai. Barely any generals were around in Japan skilled in large troop movement, and battles at that time in Japan were normally one on one. Not to mention the fact that the Mongols had better long range weapons (short bows with poisoned arrows to compensate inferior power and grenades with ceramic shell thrown by slings to terrify enemy's horses.) In ground combat, the Mongols, IMO, could have wiped the floor with the armies of Japan. But as stated, their naval ineptitude was...well..

I don't know much of military tactics, but I do know one of the reasons why Japan has never been invaded till WWII was the need for a strong naval force take take these Islands. Something that never existed in the far east till the 20th ceentury. Also the steppe calvary would find it tough going in the mountainous terrain of Japan. However, even if they won the final battle, and the kamikaze didn't strike I'm not convinced the Khan would of held onto Japan.

Wining wars is one thing, keeping control is another. A lot of the time the mongols were satisfied with tribute payments rather than conquest, remember also they were never really a completely civilized people. Therefore long term rule was never really a realistic proposition. Also, the Mongols where never competely unified, this along with other things is one of the many reasons the fell apart in the end. A lot of the time the mongol armies where made up of men that hated them and merceneries that had no real loyalty to them. So I think there are just to many things against them and it's a miracle they held toghether as long (or as short) as they did.

As I understand it, the Khans army did establish a shortlived bridgehead in Western Japan but could not sustain it. The secod attempt fared even worse. The army was not well integrated as there was on the one hand Chinese officers with some experience of dealing with water, and, on the other hand, Mongol overlords who did not and gave commands impossible to implement. As endersgame pointed out, the Samurai armies were not strong on working as a coherent unit, though.