I don't know enough about other wars to know if the American Civil War featured more multi-day battles, but will take a stab at why it did, and why others might not.
The North and South had very different advantages. The North had a numerical advantage that ran has high as 2 to 1 in the latter part, while the South had the better generals. Thus,
There were several American civil war battles where one side, typically the South got the better of the other the first day, and then the second side, the North, got the better of it on the second and subsequent days. That would be because a skillful southern general caught "half" (or so), of a Union army at a numerical disadvantage on the first day, while the North received reinforcments that turned the tide on the second day.
At the battle of Shiloh, for instance, Confederate general Albert Sydney Johnston attacked Ulysses S. Grant with 47,000 men against 33,000 for Grant. The South was winning until Johnston was shot and killed, which may have put a damper on Southern morale. Also, Grant had assembled ALL of his artillery (field artillery, siege guns, gunboats) at the Pittsburgh Landing for a "last stand" (the Union had an important advantage in this branch of service.)
Grant received reinforcements of 28,000 on the second day, his own "lost" division under General Lew Wallace (future author of Ben Hur) and three divisions of a supporting army under Don Carlos Buell, and won the battle the second day.
At the battle of Gettysburg, the Confederate infantry got much the better of it the first day against one third of the Union army, but the remaining two thirds arrived the second day and held off the Confederate attacks. On the third day, the Confederates received the belated arrival of their cavalry, which helped bolster their spirits enough to launch "Pickett's charge," and the rest is history.
What these battles had in common was a temporary Confederate advantage in the early part of the battle, and a decision in favor of the Union when they received reinforcements on the second day.
Why didn't this happen in other battles of say Marlborough? In most cases, the two sides were more evenly matched inb both numbers and generalship, and there wasn't a pattern of shifting fortunes due to the presence or absence of half or so of one side's army. The multiday battle pattern appears NOT to have anything to do with American exceptionalism.
As for the ancient multiday battles, Thermopylae was basically a siege, at least after a Greek traitor showed the Persians a "back" road over which to surround the Greeks, while the battle of Leipzig was decided by some Saxon troops deserting Napoleon and joining the Allies, reinforcing them, and "de reinforcing" Napoleon.