I think that the reason your mind pops is that you are not trying to understand what the person saying that is actually meaning.
Stating that we are no more than statistical inference is no different to saying that we are no more math - really, you can build the deepest mathematical truths on statistical inference.
The thing is that the math/s.i. that ML models do is too trivial in comparison to the math that would be required to describe a human identity. It can detect faces very well, it can calculate fifths roots very quickly, it can solve discrete problems quickly, yes, but its achievements are, in comparison to actual human understanding (and purpose), more closer to those of another tool, like a hammer, than those of humans.
edit: typos/grammar