Why shape of diamond is like diamond?
Diamond has the shape of diamond for precisely the same reason why any other crystal has the shape it has. Some crystal faces have lower surface energy than others, so the crystal grows in these directions, as if it wanted to develop these faces and avoid developing others.

This, BTW, is not determined by crystal family alone. Think of pyrite: it also has cubic unit cell, but it mostly develops the face (001) (see Miller indices) and hence makes those nice cubic crystals. On the contrary, diamond prefers the face (111), which together with its symmetry equivalents produces an octahedron. And that's why natural diamonds are typically found in this shape (not counting scratches and fractures).

Because of its low energy, the (111) face is also the hardest face of a diamond, to the point that jewelers deliberately avoid it when cutting brilliants, for it can't be polished quite as good as any other arbitrary plane.

So it goes.