Somebody else probably signed one of the forms for her. I would have to guess that Louis and Rose were not literate in English. First names are more of a problem especially for Rose. Note how her first name changes from child to child. Rose's maiden name is consistent with slight variation in spelling.
Ages are interesting too.
The ages given for Louis would place his birth in 1861 or so. His gravestone and Death Certificate place his birth in 1857. Anne and Sam, number 3 and 4 are shown with 8 and 9 previous births. Rose may have lost 5 children prior to 1893. It is not until Gussie that there is any additional indication of non surviving children.