Here in Virginia, lots of antiques stores have “Civil War bullets” available for sale, supposedly fired during the war and found years later. Once in a while, you will find other artifacts like belt buckles that were supposedly worn by the boys in blue or gray.
To what extent could Union Army artifacts be considered as the legal property of today’s US Army or Department of Defense? Are bullets a special case because since there was no expectation that they would be retrieved after being fired, they are “abandoned property” as a matter of law and can be claimed by anyone who finds them? Are belt buckles and holsters different because they were durable, or does military equipment discarded in the field either intentionally, accidentally (Private Jones accidentally dropped his canteen and couldn’t go back to get it because he would have to fall out of marching/was in a hurried retreat/etc), or when a soldier fell automatically become abandoned property as a matter of law? Does the fact that the US government did not retrieve the items after a reasonable time had passed after the war count as abandonment?
What about Confederate equipment? Does the US theoretically have a claim of right as the successor state to the Confederacy (if the is a successor state), or did any vestige of ownership pass with the Confederacy, or could it be held under US law that Confederate military equipment was owned by no one because the Confederacy was illegal under US law?
Remember that when soldiers are mustered out, they will generally be allowed to keep a certain amount of personal gear, like their uniform and so on - stuff that isn’t practical or worth it to reuse for another soldier. It becomes their personal property at that point.
At common law, things which are left lying around in a manner that demonstrates the intent of the owner to leave it there and not come back is considered abandoned, as you posit.
As a matter of practicality, US courts generally apply this rule the same way to property which wasn’t purposely left lying around, but because the owner was forced to do so - such as retreating soldiers. The exception is stuff belonging to Native American tribes which were forced off their land (and sometimes, tribes which weren’t). If you dig up a Native American artifact there’s a good chance it will be deemed the property of the tribe that previously owned it, or their successors.
As far as things like canteens, those were typically the property of individual soldiers, rather than the government.
Who owns things today also depends on where it’s found. Antiquities discovered on public land are usually the property of the finder. On the other hand, those discovered on private land may belong to the property owner rather than the finder.
The National Park Service administers most Civil War battlefields, and taking away artifacts or using metal detectors is against the law. That said, it’s only intermittently enforced, as the park rangers can’t be everywhere at once.
As to ownership of Confederate property, my understanding is that the U.S. owns abandoned or lost Confederate property such as sunken warships, military facilities, etc. The U.S. never recognized that the Confederacy was an independent nation, and in defeating it, the U.S. gained title to all of its stuff. There was a court case about that awhile back, but I can’t find reference to it online.