Is it permissible to be named with names such as "al-Abd al-Aziz" and "al-Abd al-Qadir", and does the definite article "al-" in these names refer to the same person, or is the correct name without "al-", such as "Abd al-Aziz" and "Abd al-Qadir"?

It is permissible to be named with names common to Allah and His creation, such as "'Aziz" (Mighty/Dear One) and "Qadir" (Powerful), whether they are definite or indefinite. If the word "'Abd" (servant) is added to them, such as "'Abd al-'Aziz," it becomes a name for Allah. As for "al-'Abd al-'Aziz," the name for the person is composed of the two descriptions "al-'Abd" and "al-'Aziz," and there is no harm in it, even though it is better to add servitude to the beautiful names of Allah.