Before answering the question, I would like to clarify one thing: for most purposes, [物]{もの} and [者]{もの} are not two separate words, but a single word もの which has two kanji notations depending on its meaning. This is clearer when we consider compound words such as にせもの. When someone uses the word にせもの, it is not always clear even to the speaker whether it is 偽物 or 偽者. This is because we treat にせもの as a single word, not the common pronunciation of two distinct words.
But this is just an analysis of the modern usage of the word. Is it reasonable to consider that もの was historically a single word from the beginning? I think that it is, and the best explanation might come from Occam’s razor. Assuming that we did not have a distinction between animate and inanimate things is simpler than assuming that we had two separate words for animate and inanimate things which happened to have the same pronunciation (or which used to have distinct pronunciations but were later merged). Therefore it is reasonable to assume the former unless we have evidence for the latter possibility.