Their claim to having a Judiac or Christian faith is enough to make them eligible Jew or Christian. However, this permission was granted only when the Muslims dominated Arabia back then. It shows that this permission is associated with cultural or political superiority. See the verse: 5: 5
أُحِلَّ لَكُمُ الطَّيِّبَاتُ ۖ وَطَعَامُ الَّذِينَ أُوتُوا الْكِتَابَ حِلٌّ لَّكُمْ وَطَعَامُكُمْ حِلٌّ لَّهُمْ ۖ وَالْمُحْصَنَاتُ مِنَ الْمُؤْمِنَاتِ وَالْمُحْصَنَاتُ مِنَ الَّذِينَ أُوتُوا الْكِتَابَ مِن قَبْلِكُمْ إِذَا آتَيْتُمُوهُنَّ أُجُورَهُنَّ مُحْصِنِينَ غَيْرَ مُسَافِحِينَ وَلَا مُتَّخِذِي أَخْدَانٍ ۗ وَمَن يَكْفُرْ بِالْإِيمَانِ فَقَدْ حَبِطَ عَمَلُهُ وَهُوَ فِي الْآخِرَةِ مِنَ الْخَاسِرِينَ
All pure things have now been made lawful to you. [Consequently,] the food of the People of the Book is lawful to you and yours to them. Similarly, [the bounds and limits of polytheism and monotheism have also become evident; hence] lawful to you are the chaste women of the Muslims and also the chaste women of those who were given the Book before you when you give them their dowers with the condition that you too desire chastity not lewdness nor becoming secret paramours. [However, always keep your faith pure] and [do bear in mind that] those who reject faith, all their hard work ended in vain, and in the world to come they will be among the losers.
Ghamidi saheb writes:
It should be kept in mind that the verse quoted above begins with the word اَلْيَوْم (this day). It is evident from this word that this permission had a lot to do with the fact that monotheism had become distinct from polytheism and no confusion remained between the two. It was thus expected that if Muslim men would marry among the People of the Book these women would be positively influenced by Islam. In this way, not only would there be no clash between monotheism and polytheism, but also there was a great chance that most of them would accept Islam. Consequently, Muslims today must necessarily take this aspect into consideration if they want to benefit from this permission.
https://www.javedahmedghamidi.org/#!/quran?chapter=5¶graph=5&type=Ghamidi