I really wonder if it has something to do with humans using tools, and it just works better for a tribe if most people use tools with the same "handedness", because they can share more and be more efficient. Like if hunters could swap bows when one broke, their tribe would get more meat?
No, clear "handedness" is seen in a lot of animals not just primates. You can see it easily in pets: cats tend to reach with a certain paw, dogs tend to dig with a certain rear leg over the other. Horses lead with a preferred side when possible. Similar things are observed in non-domesticated animals like wild birds and whales also. Lateral preference isn't a universal animal behavior but it's very common.
