The point of view is first person, not second person. First person narrators refer to themselves as "I" and are present in the story they tell. Second person narrators, on the other hand, address the reader as "you," as if they are the voice of the subconscious. In second person narration, the reader is made a part of the story.
Hemingway's use of point of view in this story is somewhat atypical. The "I" point of view is a way of organizing the material of the story—everything relates to the person telling the story, whose perspective controls how we understand what happens. In the story, however, the narrator seems very remote from the main problem, which is the boy's illness. The episode where the narrator goes out for a walk and shoots two birds and is happy that so many are left to shoot later is jarring in a couple of ways: one, this pleasure seems to be given equal weight with the boy's illness in the story; and two, there is a kind of emotional disconnect between the narrator's care for the boy who thinks he is dying and his lack of empathy for the birds, who exist solely to be killed. Even though the story is told from the narrator's point of view, his actual emotions remain hidden from the reader.