Many “adoptions” in the United States, especially in the 19th century and before, were informal. There was no legal paperwork or documentation created.
Family members (close or extended) or neighbors may have taken a child in to raise–perhaps because the parents were deceased or because they had too many children. Children may simply have lived with other family members long-term for one reason or another.
My great-great-grandmother, born in the 1860s, lived with her maternal grandparents for most of her childhood–evidenced by census records. She also received a bequest in the will of her maternal grandfather where her other siblings did not. The reason? I’m not certain, but I do know that her own parents had quite a few children and, for a time, her grandparents had her aged great-grandmother also living with them. It’s possible that they simply needed a little help around the house.
Another great-grandmother and her sibling were taken in by neighbors in the 1880s after their mother died and their father was able to tend to them by himself. There was no adoption.
One should still look for local court records in these cases, as sometimes there was official paperwork (either an adoption or potentially a guardianship). Just do not be surprised if there is no mention of the child or the “adoptive parents” in local court records.