Neat find!
Time to clear some issues up: yes, cane and bamboo are both members of the grass family (Poaceae). Yes, there are several canes:
Arundo donax is the giant reed that is used to make the reeds for saxiphones and clarinets, while
Phragmites communis is another reed. Both are used for arrows, although
Phragmites might be the more widely used of the two. There are certainly other reeds, bulrushes, etc, but they aren't germane here. It is possible to make a bow out of reeds, but that's not what you have here. If you want to see a bow made from reeds, pick up
The Traditional Bowyer's Bible, Volume 1
Bamboos grow all through the tropics: they're a large group of species, and a number of different species are useful for arrows. Rather fewer are useful for bows, mostly because bamboo tends to splinter under stress (I know, from having broken a friend's bamboo berimbau--a berimbau is a musical bow). The bamboo used for the bow here was definitely from a large species. It makes me think of one of the big Asian timber bamboos (
Dendrocalamus), although I can't think of a way short of using DNA technology to figure out what species it is.
I'm not an expert, but African bows tend to have a pretty distinctive way of binding the string. Rather than using loops on both ends (as here), many tribes run the string over (or through) the tip of the bow, then spiral the string down the shaft and tie it off after several turns (
example). That's not what we see here.
While I can't positively eliminate an African origin, to me the construction of the bow is Asian. There's an outside chance that the set is from south Indian (based on the shape of the arrowheads), I tend to think it's from South East Asia, somewhere between the eastern Himalayas and Borneo. This is based on the way the bow is built (typically Eurasian, with a woven, spliced bowstring and shoulder knocks) and the bow material (bamboo). Cane arrows are found all over the world, so the arrow material isn't very diagnostic.
My 0.02 cents,
F