I believe that the simple sentence that describes the relationship between the old man and the boy is
"The old man had taught the boy to fish and the boy loved him".
Santiago had taken the boy, Manolin, under his wing when the boy was only five years old. They had gone out in the boat, and Santiago remembers that the boy had very nearly been killed when he brought the fish in too green and "he nearly tore the boat to pieces". The boy clearly remembers details of that incident, but does not seem to have been afraid; he recalls Santiago "throwing (him) into the bow where the wet coiled lines were" to keep him safely out of the way. Santiago has always included Manolin in all aspects of his craft, and the boy appreciates that. His own father prefers to do things on his own, and sometimes makes the boy feel "inferior". Santiago makes the boy feel capable, and Manolin thinks the old man is "the best fisherman", far better than the "many good fishermen and some great ones" of which he knows.
Manolin is obedient to his father, but it is Santiago whom he loves. His father has forbidden the boy to fish with Santiago because he thinks the old man is no longer an effective fisherman; Manolin "must obey" because "(he) is a boy", but still, he has faith in Santiago. Despite not being allowed to fish with him anymore, Manolin looks after Santiago, making sure he has bait and food, and lovingly anticipating his needs. Realizing that the village water supply is a good distance from the old man's home, he brings him water and washing supplies, and plans to "get him another shirt and a jacket for the winter and some sort of shoes and another blanket".
Santiago and Manolin enjoy each others' company, and share a mutual respect. Their relationship is based on love, and they look out for each other like a father and a son, Santiago having taught the boy his trade when he was young, and Manolin looking after Santiago now that he is old.