Al-Aziz is The All-Mighty, the Eminent. It signifies a rare existence for whom there is intense need as well as to whom access to may be challenging.
A king is ”Aziz” (mighty and honorable): If he has all matters under his control, and if all of his subjects, young and old, venerable and disreputable, go to him for their affairs. In other words, the more needed you are, more Aziz (the mightier and more honorable) you are. If a true believer is badly needed, he is Aziz (mighty and honorable), nevertheless, he is modest.