This is indeed odd. youtube.kr is a US-based website hosted in the US. I don't see how Korea thinks it has jurisdiction over all websites using korean characters. (If they can demand that youtube.kr require real names, why can't they demand that youtube.com require that? I can log into youtube.kr with my .com account, which is definitely not using my real name. In fact, I am automatically logged in.)
If I were less cynical I would make some comment about this being an April Fool's joke... but it's probably not.
South Korea does have the power to regulate youtube.kr. This is because .kr is a country code top-level domain and the registry for .kr domains is the National Internet Development Agency of Korea (http://en.wikipedia.org/wiki/National_Internet_Development_A...). There are restrictions placed on .kr domains (http://en.wikipedia.org/wiki/.kr) such as a requirement to have a local presence in South Korea, etc.
This gives South Korea power to affect youtube.kr. South Korea has no direct power to regulate youtube.com, although it would have indirect power to regulate youtube.com in South Korea via its power to threaten Google that it can take away the youtube.kr domain.
I mean, China we can understand. That's a sixth of the world's population so even if their policies are absolutely retarded; there's some sense to Google listening to them in so much as it affects the rest of the world.
South Korea is currently ~2.5% of global GDP, China ~6%. The UK is ~3.9%. South Korea is legitimately a big deal, and writing off that market would be dumb, by the most conservative estimate its the 16th biggest economy in the world, and its internet active population is waaay above trend, so its even more important for Google than that alone would suggest.
"we have a responsibility to protect your privacy and security" is compatible with forcing users to use real names. It is not compatible with disclosing private data that the user has entrusted to you.
If I were less cynical I would make some comment about this being an April Fool's joke... but it's probably not.