One thing to keep in mind is that some who fly the Confederate flag may not necessarily be descendants of Ohioans who fought for the union during the Civil War. Instead, they are likely to be descendants of southerners who migrated northward sometime in the last 150 or so years.
Some could also be descended from some anti-war/pro-Confederate sympathizers in the north known as the “Copperheads” during the Civil War, including in Ohio.
One copperhead organization, Knights of the Golden Circle was founded in Ohio and 2 Ohio politicians such as Congressman Clement Vallandigham who was convicted by a Union military court of opposing the war and exiled to the Confederacy for the duration of the war.