Supply issues do to differing ammunition types was not new during the ACW and it certainly wasn't the last time it was an issue. The French had an absolute nightmare of it during the Franco Prussian War and had seemed to have learned their lesson by WW1, but had forgotten it again by WW2. The Germans had similar issues during WW2 as did we in Korea.
Even today in a US Ammo dump you'll find a variety of small arms munitions: 9mm, .38 (yes the USAF still has .38's so stop laughing) .45, 7.62 NATO, 5.56mm, .338 Lapua, .300 Win Mag and certainly .50. It is conceivable one Company sized unit might need to draw some of each though the most common will be 9mm, 5.56 & 7.62 NATO, that's still three different munitions.
It isn't so much the different ammo that can cause issues as during WW2 US Infantry would reasonably be expected to have .45, .30 Carbine & .30-06. It's that noone even remotely competent will mistake .30-06 for .30 carbine or .45 and issue the wrong ammo out. During the ACW w/ .71, .69, .58 & .54 in the same Infantry unit... things are a bit different. Visually there isn't a lot of difference between .58 & .54, especially in the dark chaotic moments of combat. I think the issie of multiple calibers was worse in the ACW because of the variety of weapons and the relative likeness of the packaged ammunition. All of that said there are very few battles where a battle was lost because the wrong ammunition was issued. Both frontline QM systems managed to get their act together in a fairly short time when it came to munitions.
Rations and fashions are another matter entirely. There can be no doubt which side of the aisle had the superior QM system.