Aye, I actually own 3 copies of it. (Long story)
And no, it runs on '95 through XP, although there are (slight) synchronization issues on 2K/XP, and it defaults to windowed mode on 2K/XP (easily fixable)

Vampyre_Dark got it right. The PS1 did not support a Z-buffer. They're just manually sorting the characters and the bg layers. You can see how its done with a Playstation emulator like EPSXE. Resident Evil uses one big 16-bit image for the bg with additional 8-bit sprites for foreground layers, whereas the FF series has all the background images split up into really small 8-bit tiles.

Well, also keep in mind, there's no real need for collision detection in the Final Fantasy map screens. All they really have to do is make a grid of what area is "walkable" and what area is not. They'll cut out a little area for some pipe, say it's "not walkable" and then program the image to show in front of the polygonal character if they are past a certain x or y coordinate, and behind if they are on the other side.

I mean, I don't know the technical terms, so it might be stated in other ways here, but if not, that's a jury rigging way to do it.