Apart from onomatopoeias, borrowed or transliterated words, or cognates [note], what are some English and Chinese words that happen to have the same meaning and similar pronunciations? I can think of two, Chinese "石头" and English stone, Chinese "苦力" and English coolie (note: coolie is from Hindi and Urdu, not Chinese). When I posted this message to Weibo, one user contributed Chinese "费" and English fee, another "屎" and shit, and "好诶" and hooray (which partially meets the requirement). For these linguistic coincidences and surprises, there is even a Facebook group dedicated to these amusing findings.
[note] A pair of cognates are two words that are in two different languages but descend from the same word in their common parent language. Since English or any Indo-European language and Chinese are not even in the same language family, it's difficult to say there exist any true cognates, while borrowings or loan words are abundant. An interesting case is the very old borrowing, for which cognation may be justified if the term is loosely used, is the Chinese word "蜜" ("honey") and this word in a Romance language, such as French or Spanish miel. "蜜" is said to be a loanword from Tocharian, a branch of the Indo-European language family. (The latest research may be the 2017 article The Word for ‘Honey’ in Chinese, Tocharian and Sino-Vietnamese, by K. Meier, M. Peyrot.)