I reused some of vanilla grunt Fuz files in my custom dialogue response and it doesn't show my dialogue text anymore, just the grunt sound.

Makes me wonder if dialogue text is somewhat hardcoded in Fuz or Lip files? Does anyone know?

Dialogue text is stored in the plugin, see Dialog Topics

Yes, thats what I also thought, because it makes sense, but why it doesn't say my text when I reused one of vanilla FUZ files? It says my custom dialogue text otherwise (without any FUZ).

Subtitles won't display if they're not turned on, if you're too far away from the speaker, or if the line is of a certain type (like a hit line).

If the line was unvoiced, and you had a silent voice plugin, the subtitle would be displayed even when it normally wouldn't be, as long as the line has no audio.