The data is not labelled in the same way. The child has still acquired their knowledge solely from sensory experience. How do they know to apply the spoken label "mom" to the recurring pattern in their visual data stream?
With modern data sets and labelling, so much of the problem domain is deeply hard-coded into the system. It doesn't have to learn what letters and words mean and how to identify them in a totally arbitrary* visual or audio stream. It just gets a relatively minute amount of structured data that it has an embedded understanding of what to do with.
Sure there are things like OCR and speech-to-text, but I don't think you could just run your streams through those, there's just so much subtle information loss. In order to make AI that really has a chance of reaching what humans would call intelligence, I firmly believe it has to make meaning out of some kind of raw sensory experience analogous to ours.
*Ok, not totally arbitrary, a human would not learn language from a video of a forest, and that's where the "labelling" comes in, from observing other people using language, but a child can still learn any language from just that, and the labels are often vague, inaccurate, contradictory, complex, abstract, etc. There's no master training set with the right answers, you have to decide for yourself. And humans were also capable of bootstrapping language from nothing. I just don't see modern supervised learning systems ever doing things like that.