Referring to it as the 'Civil War' didn't come into widespread use until the turn of the century. Prior to that it was officially referred to as War of Southern Rebellion. And a rebellion is what it was. There was no opposing country to declare war on, just the southern states. And President Lincoln had no need to go to congress for a declaration of war to try and quell a rebellion. He had all the authority he needed in the Militia Act.
Actually, reference to "the civil war" predates the war itself as this was the commonly used phrase to describe and warn against what both sides saw to be approaching in the years before it. The feds officially called it the "War of the Rebellion," the confederates called it variations upon the "war of northern aggression" and the sort, and early post-war writers tended to refer to it as either the "civil war" or "war between the states." Civil war has been favored over all the others in recent times if for no other reason than that it is the easiest to say and shortest to write.