In 1903, immigrants often had no papers, and the passenger lists were made up by the shipping company with the names on the tickets. My cousin came to the US in the 19teens-under another name which I know.
It seems that the fiance of a girl in her home town bought a ticket to bring his future wife to the US. But the girl died before the ticket came or the ship sailed. The family sold the ticket to my cousin's family-so it wouldn't go to waste. My cousin then came to the US under the name of the dead girl-and used her right name once she was here.
Bottom line is that there were often no papers. There was no need to go to court and change a name-you used whatever name you wanted and unless you tried to defraud someone that was fine.
So Dora whatever became Dora Goldstone so that she could pass as a sister. Then if she wanted she could revert to whatever her name was-or change to something else entirely.