Monsters very seldom enter the cultural lexicon anymore; this might be the primary example of successfully doing so this century. And the American remake is a rare example of turning out a very good, discrete version of an Asian film.
Aside from the fact that monsters just don’t grab mass audiences the way they used to, I think one reason this has caught on so well is the simplicity of the design. By nature, ‘iconic’ usually means basic. I think CGI has allowed people to create overly complicated beasties, which ultimately don’t grab the imagination the same way. The early Toho monsters are the most beloved, and hell, even the early stage original Pokemon, I think, sank into kids’ minds and hearts more than the latter, more complicated ones.