Apparently it is the custom that when a child is born into a family, no matter where in the world it takes place, the family enters the birth in the records. In this case, it is a birth certificate from the Kenyan city of Mombasa registering the birth of Barack Obama, Jr., on August 4, 1961. However, the registration is a common practice in African countries whose citizens abroad have families with foreign nationals. Obama was born in Honolulu, Hawaii, to his Ann Dunham of Wichita, Kansas, and Barack Obama, Sr., of Nyangoma-Kogelo, Kenya.
Perhaps it is an honest mistake of cultural misunderstanding--or just grasping at straws?