Can cooking or processing generate more natural sugars?
The comment you cite obviously means the second explanation. The total quantity of sugars within the apple is not considered to increase in apple sauce making. Its density, however, does increase. And the increase can be substantial, as fruit is mostly water. In fact, "table sugar" is nothing but processed (concentrated) beets or cane. 

It is possible that some production processes will create new sugars, usually by enzymatic splitting of longer carbohydrates. This is where the fructose in high fructose corn syrup comes from. So you can't assume that the second meaning will be true for all cases where you start with a fruit or vegetable. It is just true for traditional applesauce recipes. 

Whether it is possible to make jam "without added sugar" is a very murky question, as "added sugar" is not really definable in this case. 

As for the healthiness of such products, this is both off topic and unanswerable. You should ask your grandmother's doctor. He's the only one who knows what he meant and intended when he prescribed the diet.