It appears that both East Asian and Western European bows developed originally from the same bows. The bow is an old invention, old enough so that we don't accurately know when it was invented. The oldest bows that have survived date to about 8000 years ago, but pieces of older bows have been found, over 10,000 years old (and older possible finds more than 17,000 years old). Projectile points over 60,000 years old have been found that have been identified as arrow heads - if this identification is correct, the bow is extremely old. These ages give plenty of time for the bow to spread across Eurasia and Africa, and even to have reached the Americas. Given a likely common origin for East Asian and Western European bows, some similarity is to be expected.
However, they are far from identical. While most pre-modern bows are a stick with a string joining the ends, there is a lot of variation possible. The early bows that have been found are straight wooden self-bows, usually fairly long - this is essentially a description of the Medieval European longbow. Similar bows were used in Melanesia, in the Americas, in India, in Africa, and elsewhere. It's a simple design, and is likely the original design of the bow.
Chinese bows, on the other hand, are laminates of horn, wood or bamboo, and sinew embedded in glue - a composite bow. These appear to have been developed in Asia in the 2nd millenium BC, and spread to the rest of Asia, and to Europe and Africa. The classic version is a recurve-reflex bow ("recurve" means that the tips of the limbs are bent forward relative to the rest of the limb, and "reflex" means that the limbs themselves are bent/curved/angled forwards when unstrung). These bows became the standard type in much of Asia, but while they reached Europe, the wooden self bow remained popular there. These composite bows look quite different.
The classic Japanese bow is a very large recurve-reflex bow, but of laminated bamboo or bamboo/wood construction rather than the horn and sinew used in the composite bow. The amount of recurve and reflex varies from a lot to a little. Such Japanese bows are typically longer than European longbows, and are asymmetric, with the upper limb much longer than the lower limb. Again, these bows look different.
Superficially, they're all sticks + strings. However, they look different enough so that they can easily be distinguished from each other at a glance. One needs to look much more closely to distinguish between, say, European longbows and Papuan longbows.
Longbow - straight wooden self bow: https://commons.wikimedia.org/wiki/File:Englishlongbow.jpg
Asian composite bow:
strung: https://commons.wikimedia.org/wiki/File:KaiyuanBowStrung.jpg
unstrung: https://commons.wikimedia.org/wiki/File:KaiyuanBowUnstrung.jpg
Japanese bow being drawn, showing the asymmetry of the limbs: https://commons.wikimedia.org/wiki/File:YumiKai.gif
Japanese bow, unstrung (an almost straight example): https://commons.wikimedia.org/wiki/File:Antique_Japanese_(samurai)_yumi_and_ya.jpg
There are some bows that look quite different, diverging even more from the original design, that can longer be described simply as "stick + string". Examples include multiple bows (e.g., the Penobscot bow):
(multiple sticks and strings) and the modern compound bow:
(stick + string + pulleys).
_yumi_and_ya.jpg){kind=link}
