According to the information I checked online: one story is that Oden is actually a Japan-style 炖烩（hotpot) originated from the Northeast of China; another is Oden originated from Kanto region of Japan. Kanto in Chinese means 关东，so its Chinese name is 关东煮. 煮 means 'boil'.

During the Japanese colonial period in Taiwan（1895-1945），more exactly, about the end of 19th century, Oden began to spread to Taiwan via a Japanese convenience store named 罗森/Lawson,(I doubt if Lawson still remains now.) later entering Chinese Mainland from Taiwan. If so, I think there should be Oden in Korea and other Japanese colonial countries such as Malaysia and Singapore, etc.
Yeah, its Japanese name is 御田 just as what you wrote. See the picture below: