truly a civil war?

Civil conflict doesn't really have strict definitions. It can involve a group of people vying for supremacy within a state (for example, the Lebanese Civil War), it can involve the break away of a state (the Ethiopian Civil War), it is poorly differentiated from revolutionary conflicts, and may involve sides so lopsided as to constitute genocide. Even conflicts between two internationally recognized states (such as North and South Vietnam) might be better understood as civil wars while those involving fighting between people who have no internationally recognized state living entirely within the internationally recognized borders of another might not (the American Indian Wars.)

wasn't the latter technically it's own nation since it seceded?

The position of the US government was that the secession was legally invalid. Lincoln was very careful to avoid terms that would recognize it is as such, even if they were in common use. So in his speech to congress in July 1861 he said:

It might seem at first thought to be of little difference whether the present movement at the South be called "secession" or "rebellion." The movers, however, well understand the difference. At the beginning they knew they could never raise their treason to any respectable magnitude by any name which implies violation of law. They knew their people possessed as much of moral sense, as much of devotion to law and order, and as much pride in and reverence for the history and Government of their common country as any other civilized and patriotic people. They knew they could make no advancement directly in the teeth of these strong and noble sentiments. Accordingly, they commenced by an insidious debauching of the public mind. They invented an ingenious sophism, which, if conceded, was followed by perfectly logical steps through all the incidents to the complete destruction of the Union. The sophism itself is that any State of the Union may consistently with the National Constitution, and therefore lawfully and peacefully , withdraw from the Union without the consent of the Union or of any other State. The little disguise that the supposed right is to be exercised only for just cause, themselves to be the sole judge of its justice, is too thin to merit any notice.

That is, of course, a common feature of civil wars, for the governing side to deny the legitimacy of the other. I'll leave it for constitutional scholars to decide whether or not the legal grounds were strong.

Ultimately no foreign government recognized the sovereignty of the confederacy.

It's an interesting question, and one policymakers struggled with throughout the war. On the basic level, I think it has to be seen as a war of nations, rather than a civil war or counterinsurgency writ large. Confederate Nationalism was a potent force, and at the start of the war, the Confederacy exercised all the prerogatives of a sovereign state over almost all the territory it claimed to control.

The U.S., for its part, insisted that it was simply a rebellion, and that the Confederate government had no legal existence. This made those bearing arms in Confederate armies liable for treason, but it also meant that the civilians in the Confederacy were innocent until proven guilty, and their property was still theres unless there was a smoking gun that they provided aid and comfort to those bearing arms against the United States.

Lincoln undermined this position pretty quickly, though; after war finally erupted with the firing on Fort Sumter, he issued orders for a blockade of Southern ports. Recognizing blockades as a weapon of capital W War, the British issued a statement of neutrality, recognizing the Confederacy as a belligerent in a Civil War, rather than as just rebels. There was an ongoing contradiction in Northern politics at the time; Democrats who believed in the constitutionality of secession believed Southerners were not liable for treason, and yet somehow deserved the rights of U.S. citizens when their community was occupied by the Union Army. Republicans who thought secession was treason wanted to treat members of (what they considered to be) U.S. states as citizens of a foreign power.

The initial policy was still reconciliation through to 1862, however. After the 'unsecession' of West Virginia, the fall of Forts Henry and Donnelson, the conquest of Tennessee, and McClellan's approach to Richmond, it seemed like the rebellion would be put down without resorting to a war of conquest, 'a remorseless and revolutionary struggle.' However, when Robert E. Lee assumed command of the Army of Northern Virginia, and developed his counterattack outside Richmond into a counteroffensive that brought the main Confederate army onto Northern soil. Simultaneously, Confederates in Tennessee launched and offensive into Kentucky.

Rather than his own operational and managerial mistakes, Lincoln blamed McClellan and the policy of reconciliation for the failure of the 1862 Virginia campaign, and decided that harder measures would be necessary. His legal justification for the Emancipation Proclamation was that a traitor's property was treated like that of someone with no one to inherit his after his death, and that it became the state's to do with as it pleases; it was not based on subjecting the Confederacy to the laws of war between nations. Furthermore, slaves had been escaping to Union lines since the beginning of the war, and private soldiers proved increasingly hard to restrain from taking certain pragmatic measures against southern civilians, such as taking fence posts for firewood or foodstuffs without paying; in a sense, the ship had sailed on reconciliation.

Rather than encouraging the South's submission, as Lincoln thought it would, they fought harder than ever. They didn't know Reconstruction would fail to establish racial equality and leave white supremacy in place; not only was the Emancipation Proclamation by far the biggest property confiscation in American history, but it directly threatened the very existence of the South's social system.

Ultimately, the South was not defeated through counter insurgency measures; it was defeated because each of its three armies was decisively defeated. The Army of Mississippi at Vicksburg capitulated after a siege, the Army of Tennessee practically self destructed during the Nashville Campaign, and the Army of Northern Virginia was finally levered our of Richmond and Petersburg and encircled during its retreat. There had been no failure of will on the part of the Confederacy; 94% of Virginia's white population had served in the Confederate army, and other states, such as Florida, had a higher percentage; the Confederacy had taken military casualties on the same scale as the Soviet Union in WWII (military losses as % of population). The U.S. never formally acknowledged the Confederate nation, and denied it ever existed, but actions speak louder than words.