Suppose we are in spherical coordinate system and if we talk about calculating the surface area of a sphere, then first we will define a differential surface area element which will be a vector (in the direction of increasing radius) and then integrate it. Where does the unit vector go after the integration. Is it still a vector after integration? I mean we can assign a unit vector to a differential element because it is nearly a point but what about the whole surface?
Further, if we take a slice of a sphere we can have three vectors of differential surface area, so if we are asked to calculate the area of the whole slice, would we have to take magnitudes of all three components because area is a scaler, right?
In the case of surface area calculation one does not need to consider the normal vector of the area element. If we had used the vector form it would of course sum up to some vector.
That form shows up in e.g. flux calculations, where there are differential contributions $d\Phi = dA \cdot B$. Here the vector $dA$ is consumed by the scalar product.