You're bang on the money: Tamsui is a romanisation of the hokkien name, not the mandarin one. There's a few other non-mando names floating around, like Taroko (an aboriginal language).
I have a Taiwanese book for school children that includes the governments romanisation, hanzi, and some kind of extended bopomofo. While the native speaker I was learning with could read hokkien-with-characters, he couldn't do it quickly, and often had to puzzle over a certain character. In cases where they had both characters and romanisation I was able to help him on the characters he was stuck on by (badly) pronouncing the romanisation. Still, I haven't met any Taiwanese hokkien speakers at all interested in romanisation. Which is a shame - it covers all the sounds, and the language does seem to be slowly dying (the fact people don't write it probably has a lot to do with it).
Ah, so when I see a sign saying "淡水/Tamsui", the first one is in Mandarin and the second one is romanized Hokkien. I assumed they were both from the same language.
Though it seems like hokkien-with-characters is a thing (from what you say, and also from what I've seen with 4-bopomofo ruby and Wiktionary), just less prevalent. Or perhaps one can look at hanzi at the same level as romanization for Hokkien (except romanization is more prevalent), i.e. "someone else's script that kinda works for us so we use it sometimes"
I have a Taiwanese book for school children that includes the governments romanisation, hanzi, and some kind of extended bopomofo. While the native speaker I was learning with could read hokkien-with-characters, he couldn't do it quickly, and often had to puzzle over a certain character. In cases where they had both characters and romanisation I was able to help him on the characters he was stuck on by (badly) pronouncing the romanisation. Still, I haven't met any Taiwanese hokkien speakers at all interested in romanisation. Which is a shame - it covers all the sounds, and the language does seem to be slowly dying (the fact people don't write it probably has a lot to do with it).