Does a person's feeling of being honored by something—such as Allah honoring His servant by saying: "My servant," or being honored by a visit from someone—count as arrogance? And what is the meaning of "honor" in this context?

"Tashrif" (تشريف) means to elevate a status or position to a higher level. "Sharaf" (شرف) is loftiness, whether material or moral. A "sharif" (شريف) man is one who possesses high status. For a person to be "tasharraf" (تشرف) by something means that he considers it an honor for himself. It is not necessary for a "sharif" person to be arrogant. A "sharif" person may be knowledgeable and of noble lineage, or possess self-esteem, or have prestige and position, yet he is humble, respects others, and accepts the truth, unlike arrogance, which is "rejecting the truth and despising people." Arrogant individuals may exist among human beings, even though their species is honored and dignified by Allah.