Wikipedia tells me:
In Islam, the Muslim scripture, the Qur'an, is taken to represent the completion of these scriptures, and to synthesize them as God's true, final, and eternal message to humanity. Because the People of the Book recognize the God of Abraham as the one and only god, as do Muslims, and they practice revealed faiths based on divine ordinances, tolerance and autonomy is accorded to them in societies governed by sharia (Islamic divine law).
That indicates to me that "the book" refers to Qur'an. Therefore Christians and Jews are included under the umbrella of the "People of the Book" only because our sacred books are "pre-echoes" of the Qur'an itself. Given that many Christians reject the idea that the Qur'an completes the scriptures, just as many Jews reject the idea that Jesus completes the Tanakh, wouldn't we be rejecting "the book"?
On the other hand, is having an authoritative set of Scriptures from God via His prophets the essence of being a "People of the Book"? The Wikipedia article seems to say that the designation has more to do with worshiping the one, true God of Abraham as being the key determinate.