I think the "real sugar" claim is just pointing out that the soda uses cane sugar instead of corn syrup. I'm not sure if you're American, but high fructose corn syrup is absolutely ubiquitous here because of the corn lobby. It's in everything, from bread to yogurt to soda. It really does taste different than cane sugar, and because its main component, fructose, is metabolized
differently than sucrose or glucose, it has been
implicated as a contributing factor for health conditions including obesity, the metabolic syndrome, and non-alcoholic fatty liver disease. Consequently, some people actively seek out "real sugar" products. Until the last decade or so, it was really difficult to find soda sweetened with real sugar rather than high fructose corn syrup in the United States. If you've heard people raving about "Mexican Coke", this is why.
The "high fructose corn syrup in everything" phenomenon is really horrifying, and I think it's hard for many non-Americans to comprehend that when we say it's in
everything, we mean it. For example, I've heard people from other countries say that commonly available brands of sandwich bread in the US taste more like cake because of all the corn syrup.