There are a number of reasons why Amir can never fully accept Hassan as a true friend, and not just as a playmate during his hours away from school. First, Hassan and Ali are both Hazara, the lowest ethnic class in Afghanistan. Amir is constantly reminded of Hassan's social status--by classmates, people on the street and even, more discreetly, Baba himself. Considered valued members of the household, Ali and Hassan nevertheless live in a mud hut on the property, not within Baba's spacious home. Although Amir enjoys Hassan's company, it is easier for him to avoid the scrutiny of others by abandoning the Hazara boy except when it suits him. Their differences are always visible to Amir: Hassan has a harelip (and Ali is crippled) and dresses simply in the local tradition; he is illiterate and gullible; and Hassan has the distinct Mongol features that so easily separates the Hazara from the other tribes in Afghanistan. As he grows older, Amir finds himself embarrassed to be seen with Hassan in public.
Jealously becomes another reason Amir cannot think of Hassan as a friend. Amir is practically obsessed with gaining the positive attentions of Baba, and when Amir sees Baba's own attentiveness of Hassan growing stronger, Amir's jealousy strikes out at Hassan. It is this streak that moves Amir to falsely accuse Hassan of stealing, thinking Baba will never trust Hassan again. Instead, Baba forgives Hassan, and Hassan shows his own loyalty to Amir by refusing to tell Baba of Amir's transgressions against him. Amir is even jealous of the few gifts Baba gives Hassan, such as the expensive kite and the payment of the operation to medically correct the boy's harelip.
Cowardice is yet another reason Amir has for refusing to call Hassan his friend. Amir doesn't have the moral fortitude to stand up for Hassan when he is verbally attacked on the streets, nor does he have the courage to intervene when Assef rapes Hassan. It becomes easier for Amir to take the cowardly way out in his dealings with Hassan, best exemplified by his placement of his own birthday presents under Hassan's mattress.