Weapons like the bow and arrow come to mind, which was invented separately by every major civilization except for the Australian Aboriginals. Where there some intercontinental trade routes that were used that could have brought these ideas around the world?
This is a situation when we must be careful where we ascribe significance.
For instance, there's an unlimited number of ways to abstractly represent the syllable "ki".
If these all looked similar, we might want to infer something. There's nothing inherent in the sound that determines the written representation. We know Latin, Greek, and Cyrillic alphabets are all related, but if we didn't, we'd have to do some serious investigating. That's far too close for such an arbitrary design.
Weapons are a particular case where it's not at all surprising that many similar designs came about. If I need to hit someone on the head really hard, there's only so many ways to do it. There are good ways and there are bad ways. There is not a good way and a bad way to write a syllable.
Humans across the world share the same body and general physical limits. A stick is better at hitting someone than a fist. A long stick is better still, but not too long or heavy sticks, since those are too hard to swing. Rocks are better for hitting people with than sticks: they're harder, denser, and can be made sharper. Rocks are too heavy and brittle, though, to swing at people, if we bothered to take the time to make them the right shape. Or we could combine the two and make a mace! Make the rock smaller but sharper and you have an axe! There's not really any other way to go about it. If you want to maximize the force from a swinging human arm, you must increase its torque. Torque increases with with a longer lever arm radius and greater force applied, so weapons attempt to optimize this equation.
Of course, the pointy stick method only works if the other person can hit you with their pointy stick too. So we make the pointy stick longer (lances, polearms) or longer and lighter and sharper (throwing spears). If we have the material (obsidian, steel, bronze...) we can make our stick pointy for the whole length (swords, macuahuitl). Again, it's all about optimizing the physics equation.
...or we can throw our stick and rocks from a distance! Throwing's hard to do for a long time though. It require a lot of force over a very short period. We can increase our torque, as before, with an atlatl, and throw our pointy sticks farther. Or we can make something else do the throwing. Spend 30 minutes in the forest, and you'll learn that tree branches are basically springs. How many of us have idly held the bottom of a twig, bent the other end back, and flung something off? There's really no other natural material that behaves like that, and it's a fairly short conceptual jump between the finger-mounted catapult and a bow. This is one situation where there's not really more than one way to skin the cat. Find me another effective, handheld way to achieve the same results as a bow using natural materials, and then we can talk.
TL;DR: Shared weapon designs are the result of people optimizing tools for the same purpose with the same starting variables.
