In Japan, it is not uncommon for a man to take his wife's name. Traditionally, if the wife's family was more powerful then the man's family, then the man joined the wife's family and adopted the wife's family name. As the Kuchiki's are one of the four great noble families, everybody is lower rank then them, except the other four great noble family. So Koga, upon marrying his captain's daughter, joined the Kuchiki clan and changed his name.