Why do you call it overgeneralization? It’s a generalization or widening of meaning. And it’s the same process that gave us the modern meaning of “dog” (as already explained). How is this not etymology? A word has a meaning (kanjis are logographs inside an otherwise syllable script) and widens it’s meaning (the rest can also be an alphabetic script). It is closer to kanji than to Chinese hanzi because it’s a logograph inside an otherwise non logographic script.