The cat in "Once Upon a Time" symbolically supports three themes of the story.
First, the cat represents the couple's unquenchable fear. For most of the story, the couple keeps making "improvements" to their home that they think will make them more secure so they won't have to be afraid of break-ins and burglaries from the "people of another color." The cat is still able to get through the barred windows and over the extended wall. When the cat manages to get through each new security upgrade, it shows that the couple's fears cannot be assuaged by creating physical barriers.
Next, the cat symbolizes that the couple's fears are baseless. The cats in the neighborhood continually set off the burglar alarms. This represents the fact that the couple's fears are "false alarms." If anything, they are worried about the wrong thing. Instead of worrying about protecting their possessions and status, they should be worried about bringing justice and stability to their society by addressing the underlying causes of the social unrest that is wracking their community. This supports the theme that fear of "the other" is baseless; once people get to know each other, they can live in mutual respect in a win-win relationship.
At the end of the story, the cat represents wisdom and foresight. The husband assures his wife that the cat will not attempt to get over the wall with the Dragon's Teeth in place because "cats always look before they leap." This supports the theme that people must evaluate their actions in terms of those actions' future repercussions. Being short-sighted and caring only for one's own tribe at the expense of the larger society will produce tragedy in the end, as the couple finds out when their son dies by becoming enmeshed in the trap his parents created.
The cat symbolizes fear, false alarm, and foresight, reinforcing several themes in the story.