When I looked at FF5, its assembly code looked very, very strange. I believe that it's actually a statically-recompiled version of the original ROM's 65816 code. The instructions looked like they were doing very roundabout things to accomplish their task, which is what I would expect from such code.
Static recompilation means that they had a program analyze the 65816 opcodes and convert them to either C or MIPS code. The R3000 CPU in the PSX wasn't powerful enough to do straightforward interpretive emulation of the 65816. They had to do something special to get it fast enough, and it appears that static compilation was their solution.
Another big reason that I believe that the games use static recompilation is that they work exactly the same way as the originals. FF4 and FF5 still have the same bugs - have Lydia cast Dejon after Calcabrina in the dwarven castle, and it'll work just like on SNES. FF6 had some bugs fixed, but not all. The US FF6 PSX was actually derived from the Japanese PSX version rather than by recompiling the US SNES version; for example, the mosaic bug in the Skills menu was fixed in US SNES but is still broken in both the Japanese and US PSX versions. No idea about Chrono Trigger.
The GBA versions, in contrast, are rewrites of the game engine. They do not have the same bugs at all, *except* when they involve script bugs - the script code is mostly identical to the SNES originals, interpreted by the new engine. For example, the bug in FF5 where you can see a scene in world 2 where Galuf speaks even after he dies.
This is in contrast to the GBA re-releases by Capcom, who apparently used static recompilation there as well. Note how Rockman & Forte, Zelda - Triforce of the Gods, Breath of Fire and Breath of Fire 2 are exactly the same. (Unlike Zelda 3, Super Mario World wasn't ported by Capcom, and it is not the same engine.)