Someone with a through knowledge of the history of sports could write about when and where basketball first entered Taiwan. I am not that person, but I want to make a few comments about what I observed about basketball in Taiwan, and what it may say about Taiwanese culture.
Basketball is popular in Taiwan, both as a spectator sport and as a participant sport. Taiwan has its own league, and NBA games are also followed avidly. Basketball courts are set up at universities, middle schools and the like.
Although I've heard that football is the world's game, Taiwan seems to be much more fond of basketball. The easiest explanation for this is that to play football you need a rather large flat area. Most large flat areas suitable for playing football are used for growing rice in Taiwan. In the crowded cities of Taiwan, just like the crowded cities of America, basketball is the most practical sport, because it can be played in very small areas. And played it is, and often. I remember walking through parks at late at night, and seeing groups of people, from middle school students up to middle aged men, playing basketball. Midnight basketball is a neccesity in an area where 25 Celsius and 90% humidity is considered normal weather, in early spring.
There is a thing called the Sapir-Warf hypothesis, that says that the physical and cultural environment of a language inlfuences its vocabulary. The same could probably be said of sports.
Basketball, from what I have seen, is played differently, both on the streets and in league games. It is hard to tell what is real and what is subjective in a game of basketball, just like in a dance performance, but I would say that on average, that the ball is passed much more before a shot is taken, and that players very rarely set into solid positions, instead preferring a game based more on split second teamwork and extremely fluid offense.
This different style of basketball (if it is not merely a figment of my imagination) could be seen as a way that the Chinese are adopting the aggressive style of American basketball into something better suited for their culture. A fast, passing based offense depends on teammate working in tacit understanding of a changing situation, which could be seen as an outgrowth of the Chinese focus on harmony. And by avoiding the shoving and pushing drives that are common in American basketball, direct physical confrontation can be avoided. So the differing nature of Chinese basketball could be seen as a way to communicate cultural values.
On the other hand, what is easier to make? A shot while someone is sticking onto you, expecting you to dribble closer to the hoop and shoot, or a shot made from a player that was previously running around unemcumbered by having to dribble a ball? While the no-look, fast paced pass often results in a turnover, it can just as often result in an unguarded player making an open shot.
On a third hand (try to borrow one for a minute!), what is more fun and what makes better exercise, warily staring down a defender as you inch closer to the hole, and then pushing into them to get a muddy, blocked shot? Or running around freely, looking for a position from which you can make a clear and easy shot?
So the differing feel and tactics of basketball in Taiwan could be seen as a way that the game has been reinterpreted to better suit cultural values, or it could be seen as merely a change of tactics that makes sense by the games internal logic, or it could be seen as a way to increase the fun and aerobics of the sport. Any interpreation is probably equally defensible, if someone wishes to take the rather unprovable presense that Taiwanese basketball is more fluid in the first place. Personally, I side with Mencius that people everywhere enjoy the same sights and sounds, and presumably the same schoolyard amusements, and thus the difference in Taiwanese basketball is just a different way that people have found to enjoy hooping it up.