what is a purebred koi? I mean, I understand purebred dog - a particular breed of dog that's not been crossed with another breed of dog (for many generations). Are there different breeds of koi? I know there are different names for koi that have different colorations, but aren't they all just koi?
I'm not sure if the different types of koi have been around long enough to be scientifically classified differently (like dogs are) but over the last 100 years there has been a lot of very selective breeding that has gone into developing very specific traits in the various types of koi: kohaku, sanke, showa, asagi, etc etc. There has been a concerted effort through selective breeding of literally billions upon billions of fry to cultivate specific body shapes, skin qualities, thickness of the color in the scales, color patterns, etc.
You might argue that these traits are completely arbitrary and unimportant--and that is 100% fine. There are multiple levels at which to enjoy koi keeping. For most people (albeit a higher percentage in North America compared to Asia and Europe) koi are simply a pet and so (like my family dog) their body conformation, kiwa, sashi, fukurin, etc make no difference. There's absolutely nothing wrong with that.
At the same time, the beauty found in today's so-called "pond-grade" koi is a direct trickle-down result of attempting to breed koi to have all of those arbitrary traits that crazy enthusiasts are willing to pay thousands of dollars for. If it weren't for the ridiculous koi beauty contests around the world that provide the financial means for breeders to continue to improve koi in the name of competition we would still only have single color carp like those found in the wild.