FF is focused on the fantasy. So it is bound to get childish at some point. Earlier games having some characters with hyped OP powers and attacks. I think it does appear childish even with dialogues. Then again games and anime universe is like that. We treat things for what they are in that case.
If you don't like' childish' things then you can't be comfortabe with final fantasy. But I don't want us to conclude that there are different forms of entertainment for adults and those ones meant for children. If you take an example here for instance, watching cartoons, many adults really like them but other people consider them childish.
I think some of these Final Fantasy games have more of a childlike thing about them than others. You could look for instance at the "World of Final Fantasy" game- and it has a very childish story and appearance, but it has some good battle mechanics,also its kind of retro !
I have not heard anyone say the plots in FF games are childish. Maybe they were created by eleven-year-old kids but so are most phones. Chinese kids are really smart and advanced. Anyway, I think FFIX might have been the work of kids. It was excellent, actually my best of the FF series, but it did have the childish aspect.