Are the names mentioned, namely "Al-Hadi" and "Aziza," which appear on product packaging, religiously prohibited, especially since "Al-Hadi" may not be intended to refer to one of Allah's Beautiful Names, and "Aziza" may be derived from "Al-Aziz"?

There is no harm in common names that are not intended to refer to something religiously revered, such as the names of Allah or the prophets. The فقهاء (jurists) have stipulated that common names not intended to refer to something revered do not take the ruling of the revered. The deciding factor is the intention of the writer; if the intention is to refer to the revered, then it is disliked; otherwise, it is not.