It is definitely a piece of Hotchkiss sabot.
The small iron object lodged in it is a broken-off portion of the base-cup's thin wedge-shaped lip. (See a larger-caliber specimen of Hotchkiss base-cup in the photo below, showing the wedge-shaped lip.)
The ruler which you thoughtfully included in your photo (thank you!) gives us a good estimate of your sabot piece's width. It does appear to be from a 3-inch caliber Hotchkiss.
The width of the rifling-groove marks on it show that it was fired from a US 3-inch Ordnance Rifle, which had 7-groove rifling.
Regards,
Pete [P.C. George]