Here's an example. Gg-uncle John Ennis and his wife adopted four children, one girl and three boys.
1. The girl, appears on the 1880 census in the Ennis household as Carrie Hull, aged 4, and the space for relationship to the head of household was left blank. She died of croup a year later and in the newspaper announcement was referred to as "Carrie Ennis, adopted daughter of J.S. Ennis." I know who she was. Her mother and one sister had died and her two surviving siblings were living with their grandparents but her birth father was still alive at the time of her death.
2. In the 1900 census, three boys are listed as adopted - Loren [20], James [16] and Ronnie [10] - all with the surname Ennis. There is no way of knowing how long they had been in the Ennis household since there is no 1890 census and the 1895 state census named only the head of household.
By 1905, Loren had left home. Although I have no idea who his birth parents were, I have traced him through the years and he continued with the Ennis surname until his death. James and Ronnie were still in the Ennis household in 1905 but they were listed under their birth names, James Roberts and Rooney Rolzer [a gross misspelling of his true name] and were called 'boarders.'
In 1910, James and Ronnie were still living with John Ennis, James listed as a boarder and Ronnie as a grandchild. James was called 'James Roberts' and he used that name for the rest of his life. Ronnie was using the Ennis surname and he used it for the rest of his life. I have been able to identify their birth parents and, in both cases, one or both parents lived well into the 1920s or beyond and were not far away.
In summary, the girl was referred to as Ennis but she was too young to have any say in the matter. Two of the boys went by Ennis and at least one of them had family living in Wisconsin. The other boy used his birth name but, even though his parents were not more than 20 km away, he didn't name them on his WWI draft registration as next of kin; he listed a brother instead.