Most doctors ask for ID now. Even as recent as 10 years ago that was a rare practice. So, yes, it was possible to get medical services under someone else's name and even possible to execute an illegal adoption that way; I have a cousin (in his 30s now) who was "adopted" like that. My aunt was unable to have children but really wanted them, she also had a minor criminal record and wasn't well off financially so she'd never be able to legally adopt. One of her friends had an unwanted pregnancy. Aunt convinced friend to give birth assuming my aunt's name and my aunt took the baby home.