Wait… since when do we start to count babies' babbling as language? We don't know about you, but we're pretty sure that no newborn we've ever heard has had anything very complicated to communicate. That's the sneaky part of "Morning Song," though: it takes a character who seems as far from linguistic competence as possible and then discusses its ability to make language. Seem odd? Well, yes. But any time a poem figures itself as an address (from "me" to "you"), chances are that it will highlight the ways that it uses language to deliver its message. This poem is no exception. The fact that it's about a baby only makes things more interesting.
The speaker doesn't form a true bond with her child until the end of the poem when she feels they are communicating for the first time.
The baby isn't actually communicating with the speaker, regardless of what the speaker thinks.