In this context, baby is sometimes used as a proper noun to indicate a singular, specific baby. It sounds more personal than 'that baby', and has been in fairly common use for as long as I can remember in both Canada and the US (at least here on the west coast).

Oh, and you may in fact here the phrases 'sister doesn't know' or 'boss doesn't know' in some circumstances, although in those instances it would be more of a slang/affectation/colloquialism and not as common as 'baby'.