Why do we call the United States "The Union" when referring to the Civil War?

The "Union" was a reference to the "more perfect union" referred to in the Preamble of the U.S. Constitution. This phrase was itself a reference back to the less-than-perfect union under the Articles of Confederation, which explicitly defined the union of the states as "perpetual". This was the legal basis for the argument the U.S. government was making that secession was illegal (hey!) and would not be recognized as a legitimate act. "Union" wasn't just a cutesy nickname, but was literally the Constitutional basis that the U.S. was fighting for.

The Confederates argued differently, that the Constitution was not perpetually binding, under a legal argument that is usually referred to as "Compact Theory". It was a contract, or compact, made not by the people of the United States, but by thirteen states, and any of those parties to the compact could leave at any time. This was a legal theory that had never had much legal support before the war, and the federal courts had given every indication would be rejected if tested. So instead of testing it, the Confederates tried to enact their interpretation of the Constitution through a war instead of a lawsuit.

In Abraham Lincoln's Inaugural Address weeks before the war broke out, he had directly talked about this legal rejection of "Compact Theory" and the Constitutional interpretation of the "Union" at length. The whole section is worth reading in full for a good summary of the legal argument being made for the perpetual "Union" at the time, but I'll just quote the first paragraph of the relevant section here:

"I hold that in contemplation of universal law and of the Constitution the Union of these States is perpetual. Perpetuity is implied, if not expressed, in the fundamental law of all national governments. It is safe to assert that no government proper ever had a provision in its organic law for its own termination. Continue to execute all the express provisions of our National Constitution, and the Union will endure forever, it being impossible to destroy it except by some action not provided for in the instrument itself...."

This echoed James Buchanan's final State of the Union address (actually, "Message to Congress" in those days) from a few months earlier, which said in part:

"...[I]t has been claimed within the last few years that any State, whenever this shall be its sovereign will and pleasure, may secede from the Union in accordance with the Constitution and without any violation of the constitutional rights of the other members of the Confederacy; that as each became parties to the Union by the vote of its own people assembled in convention, so any one of them may retire from the Union in a similar manner by the vote of such a convention.

"In order to justify secession as a constitutional remedy, it must be on the principle that the Federal Government is a mere voluntary association of States, to be dissolved at pleasure by any one of the contracting parties. If this be so, the Confederacy is a rope of sand, to be penetrated and dissolved by the first adverse wave of public opinion in any of the States...

"It was intended to be perpetual, and not to be annulled at the pleasure of any one of the contracting parties. The old Articles of Confederation were entitled "Articles of Confederation and Perpetual Union between the States," and by the thirteenth article it is expressly declared that "the articles of this Confederation shall be inviolably observed by every State, and the Union shall be perpetual." The preamble to the Constitution of the United States, having express reference to the Articles of Confederation, recites that it was established "in order to form a more perfect union." And yet it is contended that this "more perfect union" does not include the essential attribute of perpetuity.

"But that the Union was designed to be perpetual appears conclusively from the nature and extent of the powers conferred by the Constitution on the Federal Government..."

And both these speeches echoed Andrew Jackson's "Proclamation Regarding Nullification" during the 1832-33 Nullification Crisis that almost erupted into a civil war between South Carolina and the U.S. then. South Carolina had passed a law declaring secession legal, and secession would be their next step if the U.S. wouldn't allow them to ignore a tariff they didn't like ("The Tariff of Abominations"). Jackson's speech during the crisis read in part:

"The most important among these objects, that which is placed first in rank, on which all the others rest, is "to form a more perfect Union."...I consider, then, the power to annul a law of the United States, assumed by one State, incompatible with the existence of the Union, contradicted expressly by the letter of the Constitution, unauthorized by its spirit, inconsistent with every principle on which it was founded, and destructive of the great object for which it was formed.

"The States severally have not retained their entire sovereignty. It has been shown that in becoming parts of a nation, not members of a league, they surrendered many of their essential parts of sovereignty...

"The unity of our political character...commenced with its very existence...We were the UNITED STATES under the Confederation, and the name was perpetuated and the Union rendered more perfect by the federal Constitution. In none of these stages did we consider ourselves in any other light than as forming one nation..."

And all of them were really echoing George Washington's Farewell Address in 1796, where he talked about the perpetual Union that he wished to see maintained:

"In looking forward to the moment which is intended to terminate the career of my public life, my feelings do not permit me to suspend the deep acknowledgment of that debt of gratitude which I owe to my beloved country for the many honors it has conferred upon me...

"Profoundly penetrated with this idea, I shall carry it with me to my grave, as a strong incitement to unceasing vows that heaven may continue to you the choicest tokens of its beneficence; that your union and brotherly affection may be perpetual; that the free Constitution, which is the work of your hands, may be sacredly maintained; that its administration in every department may be stamped with wisdom and virtue; that, in fine, the happiness of the people of these States, under the auspices of liberty, may be made complete by so careful a preservation and so prudent a use of this blessing...

"To the efficacy and permanency of your Union, a government for the whole is indispensable. No alliance, however strict, between the parts can be an adequate substitute; they must inevitably experience the infractions and interruptions which all alliances in all times have experienced. Sensible of this momentous truth, you have improved upon your first essay, by the adoption of a constitution of government better calculated than your former for an intimate union, and for the efficacious management of your common concerns. This government, the offspring of our own choice...containing within itself a provision for its own amendment, has a just claim to your confidence and your support. Respect for its authority, compliance with its laws, acquiescence in its measures, are duties enjoined by the fundamental maxims of true liberty. The basis of our political systems is the right of the people to make and to alter their constitutions of government. But the Constitution which at any time exists, till changed by an explicit and authentic act of the whole people, is sacredly obligatory upon all. The very idea of the power and the right of the people to establish government presupposes the duty of every individual to obey the established government."

To address you other two questions:

Considering it was the side with the legitimate US government, why don't we just call them the United States, like we do when discussing the period before 1860?

Some people do do that, but the logic behind not doing so is that the Confederate government was considered illegitimate. So calling only one side "the United States" carried with it the implication that the other side was not the U.S., as though it were a legitimately separate country. Then again, calling them the "Confederacy" can also be argued to add legitimacy to their claim as the C.S.A.

Did the US refer to itself as the Union as well before the Civil War?

As demonstrated above, yes, and there are many more examples besides. In 1901, the former Secretary of State, John W. Foster, rebutted a claim in the New York Times that the Civil War had been instrumental in altering the "United States" being considered a singular rather than plural noun. In fact, this didn't develop until several decades after the war had ended, and in Foster's study, he determined it was more of a "grammar nazi" thing of olden days rather than a result of the war. "States" was plural, so earlier presidents and politicians used the plural, so as not to sound unlearned. But they still often wanted to convey ideas of a singular national nature, and when they did, they tended to use the word "Union" instead, so they could talk about "it" and "its" people and ideas. In the late 1880s and after, the vernacular began changing.