The "Fat white thing" can be a c*ck joke, but to clear things up:
It's a Mog doll. FF always has to have Moogles or a Moogle reference, like the Cid thing, and Cait Sith by himself was too small to really be effective by himself. So, two birds one stone, Moogle reference and amp up Cait Sith.
*Mog was the Moogle who was a playable character in FFVI for those who don't know