For example today both of the Koreas consider the other one as just a part of one country which is just temporarily divided. I read that during the American Civil War the Union didn't recognize the Confederacy as a sovereign nation and that "the Union" was always a single nation. The Confederate States condidered themselves to be sovereign. So what did they consider the Union to be then? Was it just a different sovereign nation or something else?
The Confederate States of America officially regarded the United States of America as a foreign country. They believed that they had fully and legally withdrawn themselves from the Union, but they did not dispute the legitimacy of what remained of that Union. Confederate President Jefferson Davis referred to the Union as "our late associates, the Northern States" in his inaugural address on February 17, 1861.
Some of the first Confederate "diplomats" were, in fact, negotiators sent to Washington in late February of 1861 to parlay with the incoming Lincoln administration. Confederate emissaries A.B. Roman, Martin Jenkins Crawford, and John Forsyth spent several months in Washington, D.C., communicating through back-channels with the incoming U.S. Secretary of State William Seward as sectional tensions rose and an armed confrontation over the possession of Fort Sumter in Charleston Harbor loomed. Their mission - to achieve a peaceful separation of the Confederacy from the Union on Confederate terms - was almost certainly doomed to failure from the start, but in those heady days of confusion and contingency quite a lot seemed possible for both sides that was only later realized to be fantasy. It would not be the last time that the Confederate government attempted to engage on equal terms with the Lincoln administration or other organs of the Union government, though these attempts were always made difficult by the Union government's steadfast insistence that the Confederacy was neither a sovereign state nor even a legitimate rebellion. The Hampton Roads Conference of February 3, 1865, is illustrative of the futile nature of these quasi-diplomatic engagements, back-grounded as they were by fundamentally incompatible and non-negotiable demands from both sides. Here's Emory M. Thomas in The Confederate Nation 1861-1865:
Through the good offices of Francis Preston Blair, Sr., scion of a Maryland political family, [Jefferson] Davis and Lincoln agreed to a discussion of peace on February 3 aboard the steamboat River Queen in Hampton Roads. Blair spoke to Davis of a military alliance between North and South to uphold the Monroe Doctrine against France in Mexico; Lincoln, although he must have known Blair was trying to arrange a peace conference, surely had no knowledge of the Mexican scheme. Lincoln attended the conference in person accompanied by Secretary of State Seward. Significantly Davis did not attend; instead he sent Vice-President [Alexander] Stephens, Senator [Robert M.T.] Hunter, and former supreme court justice from Alabama John A. Campbell. Stephens and Hunter were Davis' political opponents. Campbell was assistant secretary of war in addition to being a would-be peace-maker. Did Davis choose his representatives because he wished them to realize the folly of believing the war might end short of independence or unconditional surrender? Perhaps. At any rate the President made independence his only demand. Lincoln's demand was reunion; and the Hampton Roads Conference quickly reached an impasse. The five men met for some time, and discussions were cordial. Yet they never could surmount the fundamental barrier erected by the two presidents.