Is it grammatically correct to use "baby" as a proper noun

We don't hear "when teaching child good manners," but we certainly hear things like "when teaching junior good manners"— as well as

why Johnny can't read

tell teacher what you saw

father knows best

when precious snowflake gets out into the real world

and so on. The baby (like junior, et al) is here an archetype, for which you may substitute any particular instance, or all instances.

Not every noun is used this way, as it must be clear we are not speaking of a specific individual but of a generic example. Mr. Burns clearly isn't meeting an actual uncle when he says he wants to

… look Uncle Fritz square in the monocle and say Nein!

I think in this usage, 'baby' is being used as a noun naming a member of the family, in the same way as 'mother' or 'father'. In the same way that we say "This is for mother/father" without any articles or possessives, we could say "This is for baby".

Having said that, I have to say that I find it quite maudlin.

I can see why such magazines would want to avoid the use of your – after all, it might not be the reader's baby. (The reader could be a grandparent, or unrelated caregiver). On the other hand, if most of the readers are the baby's parents, using an article such as a or the might read very impersonal. Maybe this is a way to settle the quandary?

By the way, I'm with you – it reads like annoying baby talk, or IDS, although I don't think it's designed to read that way.