The Cold War was not a war in the sense of a violent conflict between countries. The two main antagonists (the US and the Soviet Union) never formally fought one another. So, in that way, you could argue that it should not be called a war.
However, the Cold War truly was a war in the sense that it was a struggle for survival between two countries or two ideologies. It was not fought with weapons, but it still had major consequences for the two countries who were opposing one another. You can see this from the fact that the country that lost the Cold War (the Soviet Union) no longer exists.
So, if you define war only as a conflict in which two countries fight each other with military weapons, it was not a war. But if you define war as any conflict between nations in which there are severe consequences for the loser, it clearly was a war.