I'm going to second the recommendations of FFVI, Seiken Densetsu 3, the Suikoden series, and the Chrono series in particular. I think other people gave good explanations for how they're similar to FF games but all of them have the mixture of whimsy and serious themes that FFIX has, plus they have some similarities in terms of soundtrack, characterisation, and so on. FFIV and FFV are probably also well worth further exploration, as is Secret of Mana if you enjoy Seiken Densetsu 3.
Beyond that, you might also enjoy the Star Ocean series. Star Ocean 3 is great for the most part, although it has a very controversial ending. Star Ocean 2 is frequently considered even better.