When the word "antisemitism" was first coined, it was a characterisation of a French linguist in the 19th century named Ernst Renan, who believed that Semitic peoples were inferior. "Semitic" was, of course, a language family; Semitic peoples were the peoples who spoke languages within that family. Such languages include Hebrew (which belongs to the Canaanite branch of this family), Aramaic, Phoenician, Arabic and Akkadian (amongst others). Renan believed that Jesus was a descendant of Sumerians, whose ancestors had been moved to the land of Israel by the Assyrians, and not like the Semitic peoples amongst whom he lived.
As you can tell, it's but a short step from there to the sort of race-motivated doctrines of the later 19th century, and by the time that the word "antisemitism" became popularised (by Wilhelm Marr, specifically, at the close of the century) it was synonymous with a hatred of Jews in particular.
As to why Hitler developed alliances with non-Aryan peoples, suffice it to say that he was a pragmatist as well as an ideologue. He granted the Japanese "honorary Aryan" status, for example, which is plainly absurd if being an Aryan is in some sense biologically determined. (It isn't, for the record; race itself is a social construct.)
Within this Nazi racial hierarchy, Jews inhabited the lowest rung - almost like a master race of their own, but placed diametrically opposite all that was good and healthy in the world. The fact that the word "antisemitic" originally denoted a dislike of all Semitic peoples was irrelevant - as was the fact that the word "Semitic" originally denoted speakers of particular languages. By the time that the Nazis ruled Germany, "antisemitism" had taken on a new meaning, and friendship with other Semitic peoples was no obstruction to it.