The fish is characterized by Santiago and by its own actions.
On its own, the fish is resilient and powerful. Also, the fish is Santiago's only companion on the sea, even as it is his foe. This provides a definite role (or set of roles) for the fish as a character.
Santiago talks to the fish from the very beginning of their encounter, helping to personify the animal:
Eat it so that the point of the hook goes into your heart and kills you, he thought. Come up easy and let me put the harpoon into you. All right. Are you ready? Have you been long enough at table?
Nobility, loyalty and strength of will are all attributes that Santiago projects onto the fish. From a reader/audience perspective, the fish clearly develops a stature over the course of the book. Through the trials faced by Santiago, the narrative persistently identifies the fish's struggles as well.
Santiago and the fish are implicitly connected. Just as the fish is a "great fish", Santiago is a great man. The nobility and strength of will of the fish are mirrored in Santiago.
One effect of this parallel between Santiago and the fish can be seen in the sense of tragedy that pervades the last section of the book. Manolin's tears are for the triumph and the loss represented by both Santiago and the fish.