Why is sugar a wet ingredient?
Sugar is not really a wet ingredient, it's just treated as one in certain types of baking (i.e. cakes).

When making a cake or other "fluffy" baked good, you want a fairly small amount of gluten to be produced, otherwise you'll get a chewy texture instead, and you definitely don't want a cake to be chewy like bread.

Dissolving sugar in the water inhibits the gluten-forming proteins.  It's a method (actually, the method) of adding more liquid to the mix without getting a tough, chewy cake.  It is for this reason that sugar is considered a "wet ingredient" - because the process only works properly if you add it with the water, not with the other dry ingredients.

It's even possible for sugar to be a dry and wet ingredient in the same recipe; you might add just enough sugar to the water to get the right texture, and add more to the dry ingredients for further sweetening.