It's what's called a "benched show"- the dogs are required to be present during the entire run of the show, and when they're not being shown in the ring, they must be on display, or "benched" - in much the same way as cats in a cat show are in cages lined up on a bench that the public can see. So in that way, you can see how it would be a little easier to imagine a way to add cats to a benched dog show (although the cats at Westminster will be benched in a separate area than the dogs are).
There are only six AKC shows each year that are benched, and Westminster is the oldest and most prestigious. The Westminster Kennel Club Dog Show has been around even longer than the AKC, and now being televised each year, they keep looking for things to add to pique the public's interest. Adding cats would certainly seem to fall into that category, but it's not something I'd expect to see at any of the other, less prominent dog shows that take place on any given weekend all around the country.