Covered elsewhere rather extensively, to do a search in this forum.
But for now, short skinny is that in Government venue this Remington was just a third interation order of the "Mississippi" rifle, each succeeding contract (1855, 1863) simply incorporating the latest improvements or option requests, yet in essence the same U.S. two-band pattern (compare online for yourself).
The point is this: that as common government inventory some of the 1863 interation may have been issued from the same racks with the earlier iteration. In other words the fact that 10,000 remained until final surplus doesn't mean all 10,000 were the Remington contract. This idea is for some reason abhorrent to campaign-style reenactors and even mainstream venues, who ban the 1863. It is rejected for reenactment venue not only because of it's awkwardness in drill with 3-banders, but because the hobby has become more "effete" with improved availability of more authentically-justified U.S. pattern three-banders.
So despite the value reenactors and collectors might place on absolute provenance, as a practical matter to Mr. and Mrs. John Q. Public it's enough to know that government-issue .58 cal two-band percussion rifles saw battle in the CW. There's really no call for a hissy-fit over their occasional use at mainstream events, as long as safety concerns (no mixed types) are addressed.
It has NOT helped that these 1862/63 Remington contract rifles gained the modern-day nickname "Zuave" simply because of the 1851/55 versions that were issued to several Zuave units.
The 1863 iteration is considered by some, shooters especially, to be the ne ultra of CW muzzle-load rifle design. That original surplus inventory, once gone, fueled the type being one of the first CW muzzle-loader types to be reproduced in quantity overseas, the Italians hedging the value of these guns as shooters but sensing the profit to be made in marketing them as a common arm of the CW, supplying the needs of early reenactment infantry (history be darned).