The fact Kudara/Baekje kingdom is located in present day Korea does not mean it was populated with people having Korean culture and speaking Korean, as like you said some part of the peninsula has its own history. Some sort of Japonic language was spoken in the Korean peninsula[1] and the alliance between Baekje and some Japanese state at the time could have been because of related a culture.
In the hypothesis it wasn't the case, Japanese are indeed good at "Japanese-washing" successful people, by ignoring their foreign roots when convenient, while doing the reverse for people less successful. A pretty recent example is that of Momofuku Ando, inventor a cup ramen, born Taiwanese.
[1]https://en.m.wikipedia.org/wiki/Peninsular_Japonic