Why isn't the description "People of the Book" applied to Muslims, even though they have the Noble Qur'an?

The Muslim Ummah is the most honorable of nations, and the possessor of the most honorable book. We are people of a book, in fact, the best book. However, the term "People of the Book" (Ahl al-Kitāb) in Islamic legal usage refers to Jews and Christians, as stated in the Almighty's saying: "Lest you say: 'The Book was sent down only to two groups before us.'" The two groups meant here are the Jews and Christians. This title is not applied to Muslims because it has become specific to Jews and Christians. The "Book" referred to here means the Torah and the Gospel. Therefore, whoever embraces Islam from among the Jews and Christians is not described as being among the People of the Book in the Quranic idiom.