In the case of d⋅N=0d \cdot N = 0d⋅N=0, the ray vector is inside or parallel to the plane, which indicates there're infinite many intersections or no intersected points. It seems reasonable to just assume there's no point of intersection in this occasiion, since nothing is in view when looking exactly at the same height of a flat two dimensional surface. Not sure if there's any better way to deal with this.
You're correct that it is reasonable to just assume there is no intersection since we typically only have colors/materials defined on the face.
For larger meshes/scenes this ray would probably intersect another plane and get color from there