Why isn't area the same as surface area? Doesn't a 2D shape have a surface? In that case, is it incorrect to call the area of it surface area?
 A: Yes, unless someone has some specific shade of meaning in mind, "surface area" and "area" are the same.  I guess someone might ask, for example, "How much area does a tent cover?"  in which case, he probably wants to know how much area on the ground is under the tent, and not the actual surface area of the tent itself.  But usually, surface area is not different from area.
A: The answer to your first question is Yes, because every single object that you can imagine on a $2D$ plane has a specific surface, which is essentially the shape itself!
Regarding your second assumption I will say Yes, it is incorrect to call "area" and "surface area" referring to $2D$ objects, because the moment you start talking about a surface you imply that you are considering a 2D object "inside" the third dimension.
To make an example, let $S$ be the area of a circle of radius $r$, then you know exactly it's area by:
$$
S=\pi\cdot r^2
$$
But, when you take this object into the third dimension and "stretch" the vertexes, the now Surface area created is different because now it counts the movement inside a new dimension, even if this shape is still with height $=0$!
Numerically you can have the element $S'$ inside the third dimension, calculated by a particular integral, because your square is now considered as a function over the $xy$ plane ($z$):
$$
S'=\iint z\:\mathrm{d}x\mathrm{d}y
$$
So now the normal shape it is the same, but it does change by a correct reference inside the third dimension as the effective "Surface" of the new $3D$ object.
