A little mash up, Final Fantasy 7 was the first FF game I ever played so I've got a bunch of fond memories tied up to this game and I'm pretty glad that it was my intro to the series.
Barrett- Considering he grew up in a small coal mining town and just looks like he'd make a good rock type trainer. Plus I thought it would be cool to give him a pokeball launcher for an arm.
Tifa- she has a couple water based moves in her limit break, (the dolphin kick move was one stuck out to me for some reason) so I wanted to pair her with a water type that would also mirror her fighter personality.
Cloud- I really didn't think too hard about this one, he's got spikey hair, Jolteon's got spikey fur, seemed like a good fit (Zapdos could have worked too).
Aeris- because she was able to cultivate a little garden in the dirty slums of Midgar I wanted to pair her with a plant pokemon and I also threw Mew in there to play up the fact that she's kind of unique and special herself, being the last survivor of the Cetra race.
Sephiroth- being that he himself is the product of a biological experiment and incredibly powerful it was only right that he get Mewtwo as his personal pokemon. I picture him as sort of a dark Professor Oak.