-- sometimes babies don't immediately have names, and we need a way to refer to them
-- professionals who work around babies and in birthing tend to address both the baby and the adults by their family relationships (it's Mom, Dad, Grandad, etc. instead of George and Laura). This is presumably because that helps it keep clear what everyone's the relationships is to the baby, and what role they should play in the whole process.
-- babies aren't really people with their own unique identities when they are first born, so maybe it is kind of awkward to address them with their full adult name.
Plus saying "your baby" is potentially overly familiar and possibly confrontational. Using baby with no article makes it indeterminate - it's the idea of their baby as opposed to the actual baby itself.
In this case 'Baby' is a standin for the baby's name. I would characterize this as 'reasonably standard' for spoken language, but a little odd in written language, but could help convey a conversational tone.
I originally thought this was an example of metonymy, but I don't think it is (this is more like 'role-instead-of-name'), so I don't know what the term for this is, and I'm finding it surprisingly hard to search for. I can imagine using it for a profession or job as well, but the contexts of use seem reasonably specific.