My dog, Roman, would often eat carrots when he had the chance. We never understood why he would just eat carrots, but that was his favorite vegetable. And since his eyes were glossy to begin with, we figured that it was a healthy and helpful snack for him randomly. As for foods in general, he eats anything.
As for my cat, Wolfy, he loves to chew more than eat. He would often take a banana and chew on it, but never eat it. The same goes with a carrot. It's quite strange, but he's the type to enjoy the juices of food than the food itself. Whenever I give him wet food, he doesn't eat the food. Instead, he drinks the water substance around it. As for food in general, he eats meats. But we hardly give him human food because his stomach, or a cat's stomach in general, are a bit more sensitive than a dogs.