They don't have to assume a Japanese name. Most do, probably the legal alias they have used all their lives when speaking in Japanese. East Asian countries do not have the same attitudes to names as unique identifiers, invariant across languages as western countries do. The third generation Koreans you refer to are Zainichi Koreans who don't hold Japanese nationality.

However, the Japanese Diet has not yet passed a resolution regarding this matter [Zainich Koreans voting rights in Japanese elections] despite several attempts by a section within Liberal Democratic Party of Japan to do so, and there is considerable public and political opposition against granting voting rights to those who have not yet adopted Japanese nationality. Instead, the requirements for naturalization has been steadily lowered for Zainichi to the point that only criminal records or affiliation to North Korea would be a hindrance for naturalisation.

I'm not saying that Japan isn't very xenophobic but if they want Japanese citizenship most Zainichi Koreans can have it with relatively little difficulty.