Interesting question!

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.