North & South Korea, Vietnam, and Japan absolutely share common culture with China. Korea used Chinese characters (Hanja) before Hangul. Japanese Kanji consists of adapted Chinese characters. Pre-1900s, Vietnam used Chinese characters.

In the United States, "Asian" really means something more like "Sino". Maybe not the right word, but originally within varying degrees of the Chinese sphere of influence throughout history. In the UK, an Indian or Pakistani person would be considered Asian. This is not true in the United States.