In 1859, there wouldn't have been any form of identity required (the vast majority of people would have been too old to possibly have a birth certificate anyway). Even today, it is not illegal to give a name different from that on your birth certificate in England as long as it is not done for the purposes of fraud. You see this on the censuses a lot - people claiming to be married who aren't, changing between surnames, changing their name back to their maiden name after seperation (divorce was far too expensive for most people).
And the answer to the other question is - of course, there wasn't a sense of an 'accurate' spelling of your name among those who couldn't read or write, although some people just learnt how to write their names and not much else. She would have given the information verbally to the registrar (or, if the birth was registered by someone from the workhouse, to the workhouse staff who then passed it on), and a good faith attempt at rendering the name accurately would have been made. Easy if her name was Mary Smith or Jane Jones, but errors of course coming in for the more complicated names, which is why sometimes my Panters show up as Painters and my Stantons as Stantens or the Bonners as Bonnors, etc. (Never mind my Czech g-g-grandma whose last name is misspelt in a different way on every bit of official paper I've seen!).
It does appear there are admission and discharge records for the workhouse, which might show the mother's age and where she was living before she entered (quite possibly, she came in around the time she was going to give birth, and left as soon as she was able). Also, if she was a domestic servant, particularly if she was live-in, she would have probably have lost her job when they found out she was pregnant, and with a baby it would be difficult to get that sort of work.
