You made the statement that it has been in the baby book for 50 or so yrs. So I imagine that is close to the age you are. My oldest is that age also, so I imagine your mother is about my age. I can tell you that back then I had NO IDEA that the Birth Certificate given to me at the hospital WAS NOT the real deal. It wasn't until after my daughter had children (21 yrs ago) and we asked for the certificate with foot prints, weight, time of birth ect. They said they no longer gave those out. I said; "You mean we don't get their little foot prints"? We had to get a stamp pad and do them ourselves. We were told the actual official certificate had to be sent for.
For your mother to have both in your baby book for 50 yrs she must have known something that most at that time didn't know.
I know because when I got my working papers at 16, she brought the book with her that contained the birth certificate. She thought it was funny. I guess its possible the state accepted the hospital version. I thought it was embarrassing to have my first hair taped to the same page and if I had known how to obtain my own certified copy I would have.
In any case, when our children were born in CT we received copies of the birth certificate a few weeks later. I believe it was done as a matter or course or the hospital gave us the form to request it when we filled out the information.