The "Brothers War" idea has sort of two meanings, but I won't try to delve too deep into the first: has often been used as a way to argue that "they were all Americans" as in, Confederate and Union soldiers were Americans and therefore "brothers." There is plenty wrong with that idea and goes far more toward memory of the war than reality.

The idea, however, that you might have family loyalties split was very real, albeit limited to region. The divided regions primarily consisted of areas in the slave-holding South that also had high amounts of Union sentiment. The border states especially, Missouri, Kentucky, and Maryland had especially high possibilities of this occurring, but within states like North Carolina, Virginia, east Tennessee, and even northern Alabama, there were high rates of anti-Confederate partisans and enlistments in the Union Armies.

Certainly in the regions where there was high partisan fighting, like in Missouri and eastern Tennessee, having loyalties be divided within town and county lines was very real, and since many of the areas in Tennessee were in mountain country and more isolated, it certainly would not be out of the question to have a high percentage of relatives on one side or the other, or both. But I think most of the time when the Civil War is considered, the guerilla fighting and internal conflict within regions is typically set apart from the conventional fighting among uniformed soldiers.

So then, of the men in uniforms marching and fighting across a dozen states, were many of them fighting their kinfolk? If they were from Kentucky, there was a good possibility they were. Kentucky, really more than any other state, embodies this concept of a "Brothers War." Kentucky itself was vital strategically to both sides. With the Ohio and Mississippi Rivers as borders on two sides, with the Tennessee and Cumberland Rivers pushing into northern Alabama toward Chattanooga and to Nashville respectively, any attempt to move into Tennessee, Alabama, or Georgia was almost exclusively going to go through Kentucky first. The Union could not control the Mississippi if they did not have the Kentucky section of it. How could you resupply men further south without control of the doorway? For the Confederates the door was recognized the same way, and for this reason Confederates late in 1861 entered the state to begin building forts on the Kentucky side to prevent Union incursion. Kentuckians, however, had an agenda of their own. As a slave state, Kentucky held to the institution as long as possible, and resisted any attempts or threats to it. At the same time, the majority of Kentucky citizens did not favor secession, but wished to remain with the Union. So it has been said very accurately, Kentucky was firmly pro-slavery and pro-Union. For this reason, when war broke out, they declared themselves neutral, and officially sided with the Union when Confederates broke the neutrality and entered the state in 1861. Despite declaring their loyalty, several dozen counties in western Kentucky declared themselves to be part of the Confederacy, and elected their own Confederate governor. This is why there's a star for Kentucky on the Confederate flags.

This political division also lent itself to the individual counties. Not surprisingly, areas that held more slaves tended to send more men to the Confederacy, so this meant large areas of the Bluegrass region (in central KY) and the tobacco growing regions in western KY gave men to the Confederacy. All in all, roughly 100,000 white men joined the Union ranks from the state, and some 40,000 served in the Confederate forces. (The question of divided loyalties is limited to the white soldiers, as there were no slaves who ran away to join the Confederates, although some 25,000 African-Americans fought from Kentucky)

While not universal, it certainly was true than many dozens of counties had split loyalties that divided entire families. Famously, President Lincoln, having married into a wealthy Lexington family, had 7 brothers-in-law fight for the Confederacy, most notably Benjamin Hardin Helm, who married Mary Todd's half-sister. Lincoln offered him a position as paymaster in the Union Army in 1861, which he declined. Helm was later killed at Chickamauga in 1863, to which Lincoln responded, "I feel as David did of old, when he was told of the death of Absalom. 'Would to God I had died for thee! Oh, Absalom, my son, my son.'" Also notable was Senator and former Governor John Crittenden of Kentucky. One son, Thomas Crittenden, was a Union General, while another son, George, was a Confederate one. The senior Crittenden remained with the Union as a Representative in the US House.

There is no way to accurately figure out how many Kentuckians (or others from other states) actually had relatives on the other side. Such an endeavor would literally be impossible to figure out with any real conclusions. But county by county, it is possible in Kentucky at least, to see how many counties supplied numbers of men to both sides of the war, and looking at last names and through genealogical work, its abundantly clear that many cousins, if not necessarily brothers, actively chose opposing sides. (I know askhistorians has a policy on anecdotes and personal histories, but as purely an aside I'd like to mention that this does include my own family. Of the line of first cousins on my father's side, two went with the Union, and three with the Confederacy, from the same two counties in Western Kentucky).

We should note, too, that it sort of is a one-way street, as in, virtually no one from the North was likely to have relatives who actively sided with the Confederacy, unless they already lived there. But it was more likely, if still uncommon in the general sense, for people in the slave states to have family join the Confederacy and others join the Union. Every Confederate state, save South Carolina, provided full regiments of white soldiers for the Union Armies.

As to the last questions, really what is asked is whether Union veterans returned to live in areas that had been either Confederate states or heavily Confederate-leaning areas of non-Confederate states. Again, this was very dependent on several things, but for the most part, yes, these men returned to their homes, wherever they were. In Kentucky, particularly, the decades following the war transformed the memory of it to such a degree that is often said Kentucky didn't secede until after the war. (For more on this, see Dr. Anne Marshall, Creating a Confederate Kentucky, 2010).

So to conclude, yes, the Civil War could be said to be a "Brothers against Brother," but only in select areas where high concentrations of both sentiments existed, and this really only occurred in the slave-holding states. So while technically true, soldiers having close relatives on the opposing side was more the exception than the rule.