They have Vietnamese forms of their Chinese name. In fact, all Vietnamese names are directly translatable (AFAIK) into Chinese names. It is the same for Korean names and (mostly) Japanese names. So the Chinese Vietnamese basically used Vietnamese forms of their names.
For me, a third-generation Chinese Singaporean, my surname is essentially a romanised form of my Chinese name 曾 as spoken in the Quanzhou dialect of China. It was romanised because the language of administration in Singapore was (and still is) English. So those in the anglosphere recognised my name as “Chinese” because Chan looks familiar to English speakers as a Chinese name.
In the same way, as the language was administration in Vietnam was (and still is) Vietnamese, the Chinese Vietnamese used the Vietnamese forms of their names. Both of us would not recognise them as “Chinese” because they are written in Vietnamese!