1. People of the book to mean the Christians in verse 4:171:

O People of the Scripture, do not commit excess in your religion ... The Messiah, Jesus, the son of Mary, was but a messenger of Allah ... And do not say, "Three" ...

Who say God is three and have Jesus in their religion? - the Christians.

2. People of the book to mean the Jews in verse 4:153:

The People of the Scripture ask you to bring down to them a book from the heaven. But they had asked of Moses [even] greater than that....

Who asked Moses of prophetic signs?- The Jews.

So, The people of the book collectively means the Jews and the Christians.

the idea of Jews and Christians being called as Ahlul Kitab probably has something to do with the fact that they themselves call their Holy Book the Bible, which means the Book.

1. People of the book to mean the Christians in verse 4:171:

O People of the Scripture, do not commit excess in your religion ... The Messiah, Jesus, the son of Mary, was but a messenger of Allah ... And do not say, "Three" ...

Who say God is three and have Jesus in their religion? - the Christians.

2. People of the book to mean the Jews in verse 4:153:

The People of the Scripture ask you to bring down to them a book from the heaven. But they had asked of Moses [even] greater than that....

Who asked Moses of prophetic signs?- The Jews.

So, The people of the book collectively means the Jews and the Christians.

The idea of Jews and Christians being called as Ahlul Kitab probably has something to do with the fact that they themselves call their Holy Book the Bible, which means the Book.