I would argue that there seems to be two main reasons for this. Firstly, and simply, addressing Santiago and Manolin as the "old man" and "the boy" respectively seems to add a certain universal dimension to the tale and Santiago's struggle. Refering to these characters in this way and not using their names enforces the way that their struggles and lives are representative of deeper elements of being human.
Secondly, I would argue that these ways of refering to Santiago and Manolin stress one of the key themes of the novel, which is the disparity between youth and old age. In a sense, the novel acts as a kind of commentary on the differences and similarities between these two stages of life. Santiago, in spite of being an "old man," does show himself capable of relating to a youthful perspective. For example, he keeps up with baseball and is able to relate well to Manolin. He respects Manolin and values his youthful strength and vision. Yet at the same time, Santiago does express some reservations about youth, as is shown when he deliberately contrasts his own understanding of the sea as a woman with the younger fisherman, who refer to the sea as male. Santiago is clearly conscious of the limitations of old age, as his physical struggles against the fish and the sharks show, yet at the same time, he is able to plan another fishing voyage and continues to dream of lions, which perhaps could be symbolic of youth. Thus the reference to Santiago as the "old man" and Manolin as "the boy" thus draws attention to this aspect of the novel.