Does God's exclusivity over some of His Beautiful Names, which He has not revealed in His Book or through the tongue of His Messenger (peace be upon him), imply our ignorance of God, and is there any apparent wisdom in that?

Knowing the beautiful names of Allah and His attributes is among the means of knowing Him and affirming His Oneness. It is the foundation of faith, the bedrock of Islam, and the fruit of Ihsan (excellence in worship). The Sacred Law has encouraged "enumerating" the ninety-nine beautiful names, and "enumerating" here includes counting, memorizing, and acting upon their meanings. Scholars have maintained that Allah has other names that none of His creation has been made aware of, and these are more than a thousand names.