Most adoptions before 1900 were informal. There were no official records of the adoption--a neighboring family simply took the child into their home. The child may have had some biological connection to the family or the child may not. The child, if "adopted" at an older age may even have known who his biological parents were, or what his or her last name was. Or he might have had no idea. There were few rules because the adoptions were not done through any legal process.
The names a person used might have varied as well. I know of instances where the "adoptee" at various times used their birth mother's last name, their father's last name, and the last name of the adoptive parents. Of course, not all adoptees knew their parents' names and census takers might have assumed all children in a household had the same last name when they actually did not.