You've slightly misinterpreted the word baby in your first three examples. It means something in which one has invested time, interest, and emotion and for which one has a proprietary feeling. Speaking hyperbolically, it's as though the object of interest is like one's own child.
I thought I should add an example of the usage that didn't involve work- or job-related situations. This turned out not to be as easy as I thought. There's even a canonical political cartoon, showing Hoover turning over the problems of the Depression to Roosevelt in 1933, captioned "It's his 'baby' now":
But I finally found a story about a man who as a hobby, salvaged his car that had been in an accident, replacing the metal body with one made entirely of wood. From that story:
So, he started salvage [sic] spruce wood from construction sites, adding up
bit-by-bit to his truck every evening after work for about nine
months. So you can say it’s his baby now.
A labor of love, with a play on the nine-month "gestation" period for the "baby."