Throughout much of American history (and in other countries as well), adoption was an informal process where court or legal action was not required. Your relative may have been adopted without any legal record of that adoption taking place. The child may have been apprenticed out to a neighbor to learn a trade (sometimes generating a legal record–but not always). If the child’s parents were deceased, but had some property, there may have been a guardian appointed to oversee the property while the child was underage. In many other cases the child was simply “taken in” or perhaps spent time with a variety of families–not all of whom may have been related.
As time moves forward, formal adoptions become more common, but those records are often sealed by court order and access to those records is dependent upon state statute.
But if your ancestor was adopted in the United States in 1880, the chance of finding a record is slim. Possible, but not too likely.