I find this thread interesting, so I think I should join in as well. Pardon me if I make any mistakes
So basically we use 脂 to talk about fats originated from animals because the left radical of 脂 is a variant of 肉 (meat), correct? In my Kanji book (written in Vietnamese, which is my mother language), they also use this kind of explanation for kanji with this radical. For example, they explained 散 like "the act of scattering meat" (not sure if I translated it correctly), so the verb 散る means "to scatter".
Regarding 脂 and 油, I'm amazed to see another similarity between Japanese and Vietnamese. We also use 2 different words when talking about fats and oil, and the distribution of which word to use is pretty much the same, at least with my current knowledge.
