Hyacinth is the name of a particular plant, the three species of the genus Hyacinthus (asparagus family, Asparagaceae), Hyacinthus litwinowii, H.orientalis and H. transcaspicus, but it is easy to discover other "hyacinths": grape hyacinth (Mascari, Leopoldia and Pseudomuscari species), water hyacinth (Eichornia species), hyacinth bean (Lablab species), summer hyacinth (Ornithogalum species)... These others are quite different plants: grape hyacinth and summer hyacinth are in the asparagus family with common hyacinth but water hyacinth is in the aquatic plant family Pontideriaceae and hyacinth bean is indeed in the bean family (Fabaceae). Apparently hyacinths were so well-known that other plants were named based on reminding people of hyacinths.
Common names are so inconsistent! It seems like every plant I look up has two or three common names.
Do you call this Louisiana sagewort or white sage or... (Artemisia ludoviciana)
Common names are what we "normally" call a plant, for example when we are speaking with friends. So they are names that are familiar to speaker and listener. Since most of us spend most of our time speaking with people who live within a few miles of us, common names easily became local names, varying across the map.