Holly's cat is a constant reminder of the lack of connection she feels to those around her. For much of the story, he represents her unwillingness (or maybe her inability) to feel tied down to anyone or anything, and the fact that she won't name him further emphasizes this: "We just sort of took up by the river one day, we don't belong to each other: he's an independent, and so am I" (4.52). Holly won't claim the cat as her own because that would signify that she's putting down roots, and this is something she's clearly adverse to doing.
Near the end of the story, the cat comes to represent something slightly different. Holly sets him free on her way to the airport, but she does so by leaving him in an unfamiliar and unfriendly-looking neighborhood. When she realizes the horrible mistake she has made and tries desperately to find him, the cat symbolizes Holly's realization that she's scared about never belonging anywhere or to anyone. All of her fears come to rest in the symbol of the cat, and the fact that she doesn't find him might tell us something about her eventual fate.