I think that your "nearly identical" understates the variation in bow and arrow design. Traditional arrows vary from about 20" to over 60" in length, from about 10 grams to well over 100 grams in weight. Shafts can be cylindrical, tapered, or barrel-shaped. Arrows can be self-pointed (the end of the shaft itself is sharpened to a point), or have points of stone, bone, horn, antler, hardwood, bamboo, bronze, iron, steel. They can be fletched or unfletched. If fletched, the number of fletches varies, they vary in length, height, and shape, and the material varies. They can be nocked or un-nocked. The head can be attached directly to the shaft, or can be attached to a separate foreshaft which is, in turn, attached to the shaft. The design of heads varies greatly. Overall, I'd say that the variation in the design of arrows alone is larger than the variation in design of spears.

Bows vary from about 20" to 80" in length, in draw weight from under 20lb to well over 100lb. They can be straight, reflexed or deflexed, recurved or decurved. Cross-sectional shapes from from round, oval, D-shaped to flatbows and paddle bows. Bows can be self-bows, laminated from multiple pieces of wood, be made from multiple pieces of wood tied together, can be sinew-backed, can be cable-backed, can be composite bows made from horn and sinew on a wood/bamboo core. Some bows shoot stones rather than arrows.

All of this variation is within constraints imposed by physics, the properties of the materials, and the size of human archers. Above this variation, there is a common design. It's quite possible that the bow was invented once, and spread. It's also possible that there were multiple inventions, or at least multiple cases of the idea being turned into practice (diffusion of the idea, rather than diffusion of actual bows and arrows). Whichever is the case, there was certainly plenty of local evolution of the basic design. This is thoroughly prehistoric, and old enough so that the answer is greatly hindered by the poor survival of the organic materials used. We can say confidently that the bow has been around for over 10,000 years, and possibly for many tens of thousands of years (perhaps over 60,000 years). For discussion of this, see https://www.reddit.com/r/AskHistorians/comments/9pgvu3/where_did_the_bow_and_arrow_originate_and_how_did/ and also note link by /u/retarredroof

For a nice book showing the diversity in bow and arrow design in traditional archery: Charles E. Grayson, Mary French, Michael J. O'Brien, Traditional Archery from Six Continents: The Charles E. Grayson Collection, University of Missouri Press, 2007: https://books.google.com/books/about/Traditional_Archery_from_Six_Continents.html?id=QwmWp4hSOMIC