The Cold War was not a war in the sense that two countries fought against each other using military force until one of them gave up. Instead, the Cold War was a conflict between the United States and its allies and the Soviet Union and its allies for domination of the world. It is called a war because it was a serious conflict whose result would impact the world greatly.
The term “war” is used here to emphasize how seriously the two blocs of countries were competing. They really had the goal of destroying each other’s system of government. They were willing to use a variety of means to achieve their goals. They were even willing to go to war, as long as the war did not directly involve the US and the USSR fighting one another. In addition, the outcome of the war would really affect the world greatly because it would determine whether communism or capitalism/democracy would come to dominate the world.
The modifier “cold” is used to show that it was not a “shooting war.” The idea was that a “real” war was “hot” because there was actual military fighting. By contrast, the Cold War was “cold” because the US and the USSR never actually officially engaged in combat with one another.
This is why the period was called the Cold War. It is up to each individual to decide if this deserves to be called a war.