I haven't read much about the use of aliases during the American Civil War, but something called "bounty jumping" was indeed a practice people engaged in, and this would likely involve the use of an alias. So during the war, one could earn a bonus for voluntarily enlisting in the army (the idea being that you might get drafted anyway, so you might as well sign up, and earn the bonus you'd miss out on if drafted). Some enterprising individuals took advantage of this practice by enlisting in a unit, collecting their enlistment bounty, then deserting to do it again in a different place with a different unit. And while it wasn't that difficult to do, the penalty if/when caught could be severe. Depending on where the person was, when they did it, and who the C.O. of a unit was (and how strict they were), the penalty for bounty jumping could be anything from fines, imprisonment, to summary execution for desertion.

So, while I can't answer OP's question directly about why their ancestor used an alias during the American Civil War, it MAY have been because they were collecting enlistment bounties. It may have been something like the ancestor was bounty jumping, but they eventually just settled on a unit that had tighter controls on their soldiers, and monitored movements more closely, or just because they felt like they had a good thing going in that unit and didn't want to risk another jump. It's hard to say with any certainty, but in terms of plausible, possible explanations, this is about the best I could come up with. Using an alias would allow a person to keep enlisting in different units without drawing suspicion on a name that might have started to get flagged if a bounty jumper used it enough times.