Praise be to Allah.
Who are the Ahlul Kitab?
Ahlul Kitab (People of the Book) consist of both believers and disbelievers, as indicated in the Quran, where Allah the Almighty mentions the disbelievers among them (interpretation of the meaning):
"O people of the Scripture!: Why do you disbelieve in the verses [about Prophet Muhammad (peace be upon him)] of Allah, while you (yourselves) bear witness (to their truth)." (Al ‘Imraan, 3:70)
"Say: O people of the Scripture! Why do you stop those who believe from the path of Allah, seeking to make it crooked, while you (yourselves) are witnesses? And Allah is not unaware of what you do". (Al ‘Imran, 3:99)
And Allah, the Almighty, said about the believers among them, making clear the difference between them and the aforementioned (interpretation of the meaning):
"Not all of them are alike; a party of the people of the Scripture stand for the right, they recite the Verses of Allah during the hours of the night, prostrating themselves in prayer. They believe in Allah and the Last Day; they enjoin what is right and forbid evil and they hasten in good works; and they are among the righteous. And whatever good they do, nothing will be rejected of them; for Allah knows well those who are the pious." (Al ‘Imran, 3:113-115)
These people also believe in our Prophet (peace and blessings be upon him) and the Quran which was revealed to him as mentioned (approximate translation of meanings is):
"And there are, certainly, among the people of the Scripture, those who believe in Allah and in that which has been revealed to you, and in that which has been revealed to them, humbling themselves before Allah. They do not sell the Verses of Allah for a little price, for them is a reward from their Lord. Surely, Allah is swift in account." (Al ‘Imran, 3:199)
Are modern Christians among Ahul Kitab?
The disbelief of the disbelieving People of the Scripture, that include Jews and Christians, in our times does not expel them from being People of the Scripture. So, the rules and regulations related to them regarding permissibility of eating their food and marrying their chaste women, in the Quran and Sunnah, are still applicable to them.
And Allah knows best.