There can be many interpretations as to why the narrator focuses on the eye of the cat in Poe's "The Black Cat". Here are a few of my own interpretations.
First, Poe, himself, seems to have a fascination with the eye. The imagery of the eye is not only seen in the story "The Black Cat", but it is also depicted in the stories "The Tell-Tale Heart" (the narrator is obsessed with the eye of the old man), "The Fall of the House of Usher" ("the vacant and eye-like windows"), "The Gold Bug" (to find the treasure the left eye must be used), "Dreamland" ("Weak human eye"), and "Hop-Frog" (how Hop-Frog's eyes change throughout the story). Poe seems to know something about what the eye holds to include it in so many of his works.
Secondly, to address one possibility as to what the eye may mean for Poe is the old saying "The Eye is the window to the soul". This interpretation comes from Matthew 6:22-23 (King James:
22 The light of the body is the eye: if therefore thine eye be single, thy whole body shall be full of light.
23 But if thine eye be evil, thy whole body shall be full of darkness. If therefore the light that is in thee be darkness, how great is that darkness!
This being said, if Poe knew of the importance of the eye in detailing who a person is/was, he would surely use the eye to denote its use to define a person or the person looking into the eye of another.
As for "The Black Cat", the cat could have "seen" who the narrator really was by looking in through his eyes. Frightened by the thought of someone knowing his true self, the narrator took the eye of the cat so as not to have any more of his soul revealed.
The fury of a demon instantly possessed me. I knew myself no longer. My original soul seemed, at once, to take its flight from my body and a more than fiendish malevolence, gin-nurtured, thrilled every fibre of my frame. I took from my waistcoat-pocket a pen-knife, opened it, grasped the poor beast by the throat, and deliberately cut one of its eyes from the socket! I blush, I burn, I shudder, while I pen the damnable atrocity.
Another interpretation of why the narrator took the eye of the cat can be his simple drunken state and hate of the cat. The cat had just bit him and he retaliates by taking the eye. Simplistic and boring and elementary, but another alternative.