Yes, it is permitted and was indeed the case over time. Here, I am starting off with your premise that neither Judaism nor Christianity permit more than one wife. Imam Ahmad ibn Hanbal documented the story of a number of companions who had married Jews and Christians while they were married to other Muslim women.
حدثني أبي قال حدثنا محمد بن جعفر قال حدثنا سعيد عن قتادة أن حذيفة بن اليمان وطلحة بن عبيد الله والجارود بن المعلي وأذينة العبدي تزوج كل واحد منهم امرأة من أهل الكتاب فقال لهم عمر بن الخطاب طلقوهن فطلقوا كلهم إلا حذيفة فقال له عمر طلقها قال تشهد أنها حرام قال هي جمرة طلقها هي جمرة طلقها قال تشهد أنها حرام قال هي جمرة قال لقد علمت أنها جمرة ولكنها لي حلال فأبى أن يطلقها فلما كان بعد طلقها فقيل له ألا كنت طلقتها حين أمرك عمر قال لا كرهت أن يظن الناس أني ركبت أمرا لا ينبغي لي
NOTE. My own translation, so treat with care.
Narrated on the authority of Sa'īd ibn Qatādah that Hudhayfah ibn al-Yamān, Talha ibn 'Ubaid Allah, Al-Jārūd ibn al-Mu'alla, and Uthayna al-'Abdi each married a woman of the people of the Scripture. 'Umar ibn al-Khattāb told them to divorce them, so they all divorced them except for Hudhayfah. 'Umar told him: "Divorce her." He [Hudhayfah] asked: "Do you bear witness that she is not permissible?" He ['Umar] said: "She is a live coal." He [Hudhayfah] asked: "I know that she is a live coal but she is permissible for me." He [Hudhayfah] refused to divorce her. After a period of time, he divorced her. He was then asked: "Should you not have divorced her when 'Umar ordered you to?" He [Hudhayfah] replied: "I disliked that people might think I had done something that I should not have done [by marrying her]."
— Masā'il al-Imām Ahmad, Vol. 2, pp. 320, Hadith 949
As for whether this would be considered a violation of the non-Muslim's wife's religious rights is a matter that cannot be addressed at large due to its wide range of context.