If you can accept squash, which refers to many different species of Cucurbita, then you can certainly accept chili. Anyway, here’s another one for you: chia, from Nahuatl.
Actually, most of the common names of plants refer to multiple species. Usually, there’s no single word to refer to most of those species. The species names, if they have one, are usually a compound of some other word with the generic. Chili seems to be different in that most kinds of chilis have a distinct single name for each type.
curare (refers to the plant Strychnos toxifera as well as its poisonous product)
plantain (genus Musa; the other plantain, genus Plantago, has a different etymology)
carnauba (source of the wax)
unknown Native American languages:
While chili does refer to many species of Capsicum, each species can have many different cultivars or varieties each of which can their own name. For example, Jalapeño and Cayenne are cultivars of C. annum, Habanero and Scotch Bonnet are cultivars of C. chinense, and Piri piri and Tabasco pepper are cultivars of C. frutescens. Bell peppers are varieties of C. annum which have been bred to have no capsaicin.