Yeah, I misspoke; I should have said "where some share a common logographic script".
But I'm actually unclear about this.
The weird thing is that certain dictionaries give pronunciations of hanzi in multiple Chinese languages (e.g. Wiktionary) including "scriptless" languages.
I've also seen cases in Taiwan where e.g. a restaurant name or place name is written in hanzi and English and the romanization is clearly not using the standard Mandarin pronunciation of the term. An example of this is 淡水 which is romanized "tamsui" in Taiwan but would be "danshui" in pinyin or "tanshui" in Wade-Giles. Syllables ending with "m" are not a thing in Mandarin as far as I know, and 水 is never romanized as "sui" (though I could chalk the 水 one up to an alternate local romanization system like POJ, which I don't know anything about).
Finally, I've also seen text in Taiwan with four-bopomofo ruby. From my understanding of bopomofo; for Mandarin you won't have more than three glyphs plus a tone mark -- the initial consonant, optional medial i/u, and optional final. This also gives me the impression that Min Nan can be written in hanzi. Unless Taiwanese Mandarin differs from Mandarin significantly in pronunciation (which may be the case, I don't know).
All this leads me to believe that other Chinese languages can use hanzi, but do so less often, perhaps? This is something I've always been confused about but never really looked into it.