Ahl al-Kitab (people of the Book) is the name of the people who were given the scripture by Allah, regardless of what their current beliefs are.
There are believers among them, who accepted Islam:
وإن من أهل الكتاب لمن يؤمن بالله وما أنزل إليكم وما أنزل إليهم خاشعين لله لا يشترون بآيات الله ثمنا قليلا أولئك لهم أجرهم عند ربهم إن الله سريع الحساب
And indeed, among the People of the Scripture are those who believe in Allah and what was revealed to you and what was revealed to them, [being] humbly submissive to Allah . They do not exchange the verses of Allah for a small price. Those will have their reward with their Lord. Indeed, Allah is swift in account.
— Quran 3:199
And there are disbelievers among them:
إن الذين كفروا من أهل الكتاب والمشركين في نار جهنم خالدين فيها
Indeed, they who disbelieved among the People of the scripture and the polytheists will be in the fire of Hell, abiding eternally therein.
— Quran 98:6
Trinitarian christians are Ahl al-Kitab, because the Quran has addressed them as such:
يا أهل الكتاب لا تغلوا في دينكم ولا تقولوا على الله إلا الحق إنما المسيح عيسى ابن مريم رسول الله وكلمته ألقاها إلى مريم وروح منه فآمنوا بالله ورسله ولا تقولوا ثلاثة
O People of the Scripture, do not commit excess in your religion or say about Allah except the truth. The Messiah, Jesus, the son of Mary, was but a messenger of Allah and His word which He directed to Mary and a soul [created at a command] from Him. So believe in Allah and His messengers. And do not say, "Three";
— Quran 4:171
Though they are obviously disbelievers because they reject the Messenger of Allah: Muhammad (ﷺ) and the Quran (4:150-151) and they do shirk (5:72).
According to the majority of the schools of thought, the permission of eating their food and marrying their women stands regardless of their disbelief, because 5:5 itself mentions their Kufr and fate and does not exclude them from the concession:
وطعام الذين أوتوا الكتاب حل لكم ... والمحصنات من الذين أوتوا الكتاب من قبلكم ... ومن يكفر بالإيمان فقد حبط عمله وهو في الآخرة من الخاسرين
The food of those who were given the Scripture is lawful for you ... [and lawful are] chaste women from among those who were given the Scripture before you ... And whoever denies the faith - his work has become worthless, and he, in the Hereafter, will be among the losers.
— Quran 5:5
A discussion of this may be found in Tafsirs of 2:221 and 5:5, example Tafsir Qurtubi on 5:5 and 2:221 and Tafsir Jassas on 5:5 and 2:221.