"Chinese" in English comes from Portuguese ("chinês"), who were some of the earliest Europeans there. -ês and -ese both come from the same Latin origin.
I hazard that words like Cantonese, etc., came either via the same route, or by back-formation to match the existing wider term "Chinese".
Interestingly, "Japanner" used to be another word for a Japanese person.
I have faced questions from Chinese people about why only Asians receive an -ese suffix in English.