The progression of sounds that a baby learns is well understood, especially the physiological aspects of language development and acquisition.
If we accept as a premise that cultures would tend to assign words most relevant to babies according to their physiological capability, then it becomes obvious why words for mother, father, and even facial features would be strongly similar across cultures. And why the similarity will slowly diminish as the meaning of words becomes less important, functionally, for communication with a child.
And especially for simple words the phonetics would have very little to do with cognitive development (and evolutionarily-dictated language models), but be controlled almost entirely by a baby's physical constraints--e.g. huge tongue in a tiny mouth, poor motor control, etc.