Along with discipline ("no girlfriends until you're married!") I think it also has to do with committing the child to something at a very young age. I mean, what kid knows they want to be a professional gymnast when they're 5? But my understanding is that's about when they start hardcore training for it.
For me, I started playing piano when I was 6. Actually, I know I started way before that (have vague memories of it), but the teacher ended up recommending for me to wait a few years because I was very fidgety and not really able to just sit there and play. I would then take piano lessons more or less continuously (though some breaks whenever we moved to a new city) until I graduated from high school. Did I actively "choose" to play piano? I don't think so, though I certainly am glad that I did stick with it now.
As already mentioned by others, I think part of it is the difference in Western versus Eastern (or at least Asian) philosophy with regards to kids. Western philosophy is more about the kid going out and figuring out his own destiny, whereas Eastern philosophy is more about the kid helping the family, bringing honor to the family name, etc. Thus parents see what the kids do sort of as an extension of themselves, just as someone here view their BMW or Mercedes or Ferrari. Having a kid that can do X is a status symbol to humblebrag to friends about for "face" in the same way someone here might pull up in a fancy car ostensibly without making a big deal about it (but of course, hoping everyone will notice).
Also, Eastern (again, or at least Asian) society highly value education, so kids are in highly structured academic programs from a very young age. School doesn't get out until near dinner (rather than mid-afternoon in the US), and then there's prep classes to take in the evening, and any "free time" is supposed to be used for studying, etc. It's very different from the Western philosophy of letting the kids run off and play after school is over to discover the world for themselves.
One reason why piano might be particularly competitive is that things Western are considered high status or culture. Just being able to speak English well is considered a sign of a good upbringing -- that your parents has enough means to send you to English prep classes, etc. So I'm not surprised about the piano since it's a Western instrument.
I'd be curious as to how well those students do when they actually get to a Western conservatory. A lot of the discipline is imposed on students from without, i.e. from parents and societal pressures, so how do they fare once they're in an environment without those pressures and they must have the discipline on their own? I've heard many stories about kids rebelling once in that environment because they don't really know how to handle the personal freedom that they're given; while in Western culture the stormy years are likely the adolescence of middle school and high school, for Asian culture it's likely in the late teens/early twenties when the kid decides to pursue a different career than what the parents wanted, or date a blondie, or whatever.