Why is the hippocampus considered to be a cortical structure but not the amygdala?
You can't distinguish cortical from deep structures from a picture like the one you reference in Gray's Clinical Neuroanatomy. The hippocampus is a cortical structure because its cell bodies are in the outer layers of the cerebrum. The amygdala is a deep structure because its cell bodies are in nuclei. I'm not sure if your textbook has stained sections, but if it does, that should help clarify the difference between the two structures.

In this sagittal section you can see both. Notice the hippocampus (at both ends) follows the bumps and grooves of the cortex where the amygdala is a circular/almond shaped cluster of cells.