When reading the Qur’an it is important to keep in mind that a particular issue may be discussed over several chapters or several verses in the same chapter. Allah (SWT) gives a warning not to study any verse of the Qur’an in isolation as it will fail to deliver the correct interpretation of such verses.
The verse in question is:
“O you who have believed, do not take the Jews and the Christians as allies. They are [in fact] allies of one another. And whoever is an ally to them among you – then indeed, he is [one] of them. Indeed, Allah guides not the wrongdoing people.” (5:51)
All throughout history, Christians and Jews have never been allies of each other, even when the Qur’an was revealed. Muslim men are permitted to marry chaste women who are from the people of the book (5:5) i.e., Christians and Jews. It would be a contradiction if Allah said not to take them as a friends yet Muslims are permitted to marry them. Allah says: “If it (i.e., the Qur’an) had been from [any] other than Allah, they would have found within it much contradiction”. (4:82)
The Qur’an explicitly identifies a community whose hearts will display great hatred towards Muslims:
You will surely find the most intense of the people in animosity toward the believers [to be] the Jews and those who associate others with Allah; and you will find the nearest of them in affection to the believers those who say, “We are Christians.” That is because among them are priests and monks and because they are not arrogant. (5:82)
We also see in the above verse that Christians will be nearest in affection to Muslims. Muslim men are permitted to marry chaste Christian women.
When considering all of the above, it is made abundantly clear that the verse in question is not referring to all Christians and Jews. We have now arrived at the moment when we can ask the question: Which Christians and Jews are allies of each other?
An alliance eventually appeared in the world in the form of the Judeo-Christian alliance which brought the State of Israel into existence. The Jews hatred towards Muslims was manifest in the life-time of prophet Muhammad (pbuh) and is on full display once again after the creation of the Zionist Movement. Even then, not all Christians and Jews are allies of each other, especially among the general population in Israel.
Muslims are prohibited from becoming the allies of the Judeo-Christian Zionist alliance.
This also includes prohibiting Muslims from joinig NATO, the UN and such. Not only are Muslims prohibited from being their allies, but such Muslims would be recognised to have become a part of them and lost their Islam (5:51).