"Natural", in this context, is a weasel word that's whose key uses include making people feel more comfortable about buying junk food, so that junk food manufacturers can sell more junk food.
Chimpanzees, for example do eat a lot of fruit. I believe it's their primary source of calories. But the wild fruits that chimps eat are a very different beast from how humans in wealthier countries consume fruit. The fruits they typically eat haven't had their sugar content dramatically increased through centuries and millennia of selective breeding. They haven't been turned into juice, which removes all the fiber and essentially renders them a nutritional equivalent of Coca-Cola. They haven't been dried, which concentrates the sugar and increases the glycemic load. They haven't had extra sugar added as a ("natural"!) preservative in order to maximize the shelf life. etc. etc.
All that aside, though, no, I'm pretty sure there hasn't been any compelling evidence to indicate that your body can somehow tell whether the sugar in your food was produced in situ or extracted from some other plant and then added to what you're eating. To it, C12H22O11 is C12H22O11. There is some stuff suggesting that processing affects how much sugar is extracted by your digestive system, though. It's not able to break down the food and get at its contents quite as efficiently when the food hasn't been mechanically ground up or macerated first, and your teeth are unlikely to grind it up quite so finely. In a nutshell, sugar that's inside a plant's cells is going to be less available (and, to that extent, "have fewer calories") than sugar that's on the outside of the cells.