Great question! The answer is a bit complicated. For background, Civil War historians are a bit divided as to just how much soldiers in the conflict would have cared about things like preserving the Union or emancipation. If you're interested in a deeper dive into the debate about that, I recommend Gary W. Gallagher's The Civil War lecture series for The Great Courses on Audible, specifically Lecture 7: The Common Soldier.

On the one hand, once secession happened, there was a massive sea change on opinion about a abolition in the North. Before 1861, abolitionists were a minority in the North. It's true that Northerners generally did not like slavery and wanted to curtail its expansion, but that isn't the same thing as calling for abolition where it already existed. Lincoln was elected on exactly this platform. After secession, however, historian James McPherson argues that there was a new consensus developing that slavery was tearing the Union apart and had to be destroyed. As he writes in Battle Cry of Freedom:

"By midsummer 1862 all but the most conservative of Republicans had come to a similar conclusion. “You can form no conception of the change of opinion here as to the Negro question,” wrote Senator John Sherman in August to his brother the general. “I am prepared for one to meet the broad issue of universal emancipation.” A conservative Boston newspaper conceded that “the great phenomenon of the year is the terrible intensity which this [emancipation] resolution has acquired. A year ago men might have faltered at the thought of proceeding to this extremity, [but now] they are in great measure prepared for it.” (McPherson, Battle Cry of Freedom, p. 496)

This, combined with the rampant popularity among Union soldiers of marching tunes which explicitly reference abolition, such Battle Cry of Freedom and Battle Hymn of The Republic*, lends credence to the idea that many Union soldiers absolutely did care. I would also point out that there was, if nothing else, a significant part of the Union army that would have cared about emancipation more than anything else: the black Americans serving in the United States Colored Troops, which comprised 10% of the total Union force by the end of the war.

On the other hand, there is also evidence that many soldiers didn't care much about it at all, and that if there was any ideological motivation for them, it was the preservation of the Union. After all, if you look at how emancipation developed as a major war aim, it started out with Union officers making the decision to confiscate slaves from their masters as the US army advanced. Slaves were a key asset for the Confederate war effort. They served as cooks, teamsters, and manual laborers for the Confederate army. Naturally, anything the Union army could do to take away this advantage, they would. This is the reason the Emancipation Proclamation was specifically aimed at states that had seceded, not Kentucky, Missouri, Delaware, or West Virginia. Emancipation was a war goal that was entirely secondary to the preservation of the Union.

And we don't think about it much today, but the sanctity of the Union was absolutely something that would stir people to armed conflict. One of my favorite episodes involves southern Unionists at the outset of the war:

"On April 18, a stagecoach carrying the mail and a few passengers arrived in Johnson County, Tennessee, from Abingdon, Virginia, some thirty miles away. Two of the men on the stagecoach were from Virginia, and they checked into a Taylorsville hotel owned by Samuel Northington. That evening, the men caused a commotion in the street by shouting and waving a Confederate flag in loud celebration of the recent secession of Virginia. Northington, a Unionist, cautioned them that Tennessee had not seceded and many townspeople did not support the Southern Confederacy. The men took offense and replied they had a right to celebrate and furthermore did not care what Unionists thought. That evening, Northington, along with his son Hector and fellow Unionist Joseph Wagner, agreed they would take the flag, by force if necessary, if the men displayed it again. The next morning, as the Virginians emerged from their rooms, they prepared to ride through town waving the Confederate flag. As they mounted their horses, the three Unionists appeared with shotguns and demanded the flag be handed over. One of the men began to swear at the elder Northington and challenged him to try to take it. To that end, Northington began shooting several holes in the flag. Caught off guard, the men handed over what was left of the flag and quickly rode out of town." (Melanie Storie, The Dreaded 13th Tennessee Union Cavalry, p. 24)

Bear in mind that these were Southerners. Their neighbors were slaveholders, and they were generally very much racists. That book describes these same men, after enlisting in the Union army, carrying walking sticks with them whilst encamped near Nashville to beat black people who wouldn't yield the sidewalk (Storie, p. 73). They cared quite a bit about the Union, and were at best ambivalent towards emancipation.

I'd say that it depended on which particular soldiers you were talking about. So, yes, many soldiers probably did care about emancipation, but many also did not. As I said, this is a matter of much discussion among historians.

There is some difficulty in answering because the kinds of sources that could most directly answer this question, like polling data, simply do not appear during this period. But there is plenty to give a general sense, some of which was covered by u/JohnBrownReloaded. There’s a few pieces of particular interest here that I think give a decent impression of what a Union soldier might have thought.

First, there is some need to discuss conscription, which did not appear in any real sense for American citizens until the Civil War. There were some efforts to conscript during the War of 1812, but these efforts were opposed and the war ended before national conscription could begin. Both Confederate and Union forces rather early in the war created a conscription system, but both faced many issues and on the Union side it is estimated that no more than 10% of the army was ever composed of draftees or those who were paid to “substitute” for a draftee. Thus there were some for whom involvement in the war had no meaning other than being drafted or paid to fill someone else’s draft, but these appear to be (for the Union side, at least) a relatively small group. Altogether, this group wouldn’t have ever been large enough to impact what the average soldier would have thought of the war, though it did impact the broader public perception and lead to some charges that the Civil War was a “rich man’s war, but a poor man’s fight.”

Second is that the perception would matter a lot based on where and when you are talking about, even within the narrow frame of the war. Abolitionist sentiment in the North was fairly limited, though the limited view was not uncommon. A large number in the North supported a kind of “compromise” position with slaveholders, wherein slavery would be maintained where it existed already but would not be allowed to expand into any new territories. The more radical abolitionists, who wanted a full elimination of slavery in all places, were few a far between even in the North during the years leading to the war. The major change was from John Brown and his raid at Harper’s Ferry. In the week following the trial, almost 250 publications reported on the raid with most asserting that at least 200 men were involved in the raid at Harper’s Ferry (the real number was vastly lower). Public opinion of Brown was, at first, sharply negative even among abolitionists in the North and most condemned Brown as a fringe radical.

However Brown’s reputation was salvaged by a number of factors. First, several notable intellectuals came to the defence of Brown. Wendell Phillips, William Lloyd Garrison, Henry David Thoreau, and Ralph Waldo Emerson all spoke in praise of Brown and his actions in the days following his raid and trial, and the Boston Transcendentalist movement altogether wrote frequently enough in support to create James Redpath’s collection, Echoes of Harper’s Ferry, that featured numerous letters and speeches by some of the most influential Northern intellectuals and abolitionists praising John Brown. This created a fairly large regional divide: while the Boston intellectual community strongly supported Brown, that of Baltimore took the near opposite view. Baltimore was a hotbed of Southern sympathy for the Union, with pro-Confederate riots breaking out in April of 1861. The “copperhead” faction pushed at times for allowing the South to secede, for not supporting the succession but not using force to combat it, or for a mere re-unification without pushing abolitionist positions. Geography mattered quite a bit: Baltimore was scarcely far from the border and the Confederate capital at Richmond, Boston much further inland and closer to the various religious communities that most strongly opposed slavery.

During the war, various regiments marched to first “John Brown’s Body” and later to the “Battle Hymn of the Republic,” both set to the tune of an older devotional “Oh, Brother.” “John Brown’s Body” was, according to the most reliable accounts, composed by the “Tiger” Battalion of the Massachusetts Militia, and in public record it was first performed at a flag raising ceremony on May 12, 1861 at Fort Warren, near Boston. Abolitionist Julia Ward Howe published her “Battle Hymn of the Republic” in February of 1862. Howe lived for most of her married life in Boston, though she also maintained a residence in Rhode Island. The earlier “John Brown’s Body” is not explicitly abolitionist, save for the implication of citing John Brown as an inspiration for fighting the war, but “Battle Hymn of the Republic” very much identifies the war as a divinely inspired conflict with abolitionist purpose, the final lines being: “In the beauty of the lilies, Christ was borne across the sea / With a glory in his bosom that transfigures you and me / As he died to make men holy, let us die to make men free / While God is marching on” and previous lines making allusions to biblical conflicts casting the South and slaveholders as the serpent in the garden.

Both songs, while associated with the Union, were even more associated with Boston and its abolitionist community, which itself was associated with the Transcendentalists and the Quakers. There was thus an important regional divide: the more interior states of New England, particularly those of Massachusetts, Pennsylvania, and Rhode Island that had long maintained active religious and philosophical communities opposed to slavery were more willing to view the Civil War as being a conflict to end slavery in the South. Conversely, states near the border like Maryland and Kentucky viewed the central issue as preserving the Union. This is in not altogether unexpected, given many border states themselves allowed slavery during the Civil War, but it is a useful distinction to make.

Nonetheless, whatever the average soldier identified as the main cause early in the war, by the late stages it would have been pretty clear that the abolition of slavery was going to be a consequence of Union victory. Whether this became a view of the reason for the war is less clear, but as JohnBrownReloaded noted there is a pretty obvious shift in the way the war is discussed and in particular the aims of abolition. I would note here however that many historians take a different view and place abolitionism as more central even before the war, see Thomas Fleming’s A Disease in the Public Mind for example. But as a general conclusion, I would say that the average Union soldier’s views on the purpose and cause of the war would have been dictated mostly by where they lived at the onset of the war and moved more into abolitionism as the war progressed and Union victory seemed more certain.