Different hospitals will have different methods to identify mothers and their babies (with the aim of not mixing them up!) The most common method used in delivery suites is placing two identification bracelets on the baby's wrist(s) and / or ankle(s) as soon as they are born. If you are in a birth centre, this will usually be done when the baby is weighed and measured (or not at all if you go home within a few hours on the early discharge program). Of course if you have your baby at home, labelling is not necessary at all.
The bracelet will include the mother's name, the baby's date of birth, sometimes their time of birth and their sex (male or female). It may also include the baby's and mother's medical record numbers. Twins (or more) will also be identified in birth order, for example 'twin 1' and 'twin 2'. You may be asked to read, and verify the details as being correct, before they are placed on your baby.
In Australia, babies in hospitals will always be identified as "Baby of (or B/O)....." and then the mother's name. The mother will be registered in the hospital as the name that is on her Medicare Card (in Australia). If you are registered under your own name (or maiden name), for example "Jane Smith", then the baby will be labelled "B/O Jane Smith". This is standard procedure, being the only way the hospital can identify a link with that baby and their mother.
If you intend to give your baby a different surname (for example your married name or your partner's name) this is done when you submit your application to the Department of Births, Deaths and Marriages in your state to register the birth. Your hospital or caregiver will give the forms for this to you within days of the birth.
The identification bracelets are checked again by the caregiver on arrival to the postnatal ward and are removed when the baby is discharged from the hospital. You may wish to keep these as a memento.