3rd generation Koreans are still considered foreigners (and they must assume a Japanese name, and oddly none the less considered foreign).
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.
It just seems incongruous to me that someone who is third or fourth generation born in Japan could be considered 'foreign'. It's not as if the parents were 'illegally' in Japan. (anyone illegal is pretty much promptly sent home --with few exceptions of people involved with crime syndicates).
It's not even ironic that zainichi just means someone 'residing/staying' temporarily, ie. not considered permanent -but they've been born and lived in Japan all their lives, as well as their parents and grandparents.
It seems incongruous to me as well but it's not my country so my opinion is of at most academic interest. I'm not that interested in the literal meaning or etymology of words. In the same way that Charlize Theron is not an African-American but Morgan Freeman is, Zainichi Korean means persons of Korean nationality who have been born and raised in Japan, not Koreans resident in Japan who were born in Korea.
Charlize Theron is South African American, nationality-wise. In terms of Continental designation, South Africans of any ethnicity (Indian, European, African, ancestry call themselves 'African'. It's only outside Africa that it might seem odd. I guess the flipside is how many/most? Egyptians consider the fact they're in Africa incidental.
Zainichi people of Korean ancestry are not Korean nationals. Korea (South) can't call them up to serve in the military for their compulsory army duty. It's like saying 4th generation Americans of Italian or Mexican descent aren't 'Americans' if their parents were not officially naturalized Americans. Zainichi for this intent means 'guest' Korean.
rean passports. They are exempt from military service unless they move to Korea. Connotations and denotations of language change, zainichi means temporary and chosen means korea but zainichi chosen refers to born and raised on Japan people with Korean citizenship or an entitlement to same, not people studying abroad or similar.
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.