From what I can tell, the massive success of Final Fantasy VII is to thank for making this era what it was. Square had built up an empire and name for itself as a serious competitor in the RPG space on the SFC/SNES, and their breakout title on PS1 was the moment it all came to a head.
With all the extra cash to splash from that one success, they did what we all wish EVERY COMPANY would do: they invested that money back into making even more great games. Lots of cool weird stuff, experimental takes on nearly every genre, all using Final Fantasy as a backbone to help sell units in a pinch! Good examples would be Brave Fencer Musashi with its Final Fantasy VIII demo and Ehrgiez with playable Cloud and Tifa!
FFVII even got a special release in Japan that allowed you to see high quality versions of the CG assets for the game.
Square was very lucky that Dragon Quest took so long to come to PS1. They were able to steal the hearts and minds of a whole generation as a result, leveraging the worldwide success of the PS1, and building the Final Fantasy brand in the process. By the end of the generation, Final Fantasy’s 2, 4, 5, 6, 7, 8, and 9 all made it to the console! Not to mention Tactics and spin offs by the dozens. It was a time to be alive, and possibly the best place and time to be an RPG fan.