He phrased it wrong. What he probably meant was: The first year in the baby's life is – well – the first year, not the zeroth.
It doesn't matter much, because we will never reconcile the "I'm counting objects" and the "I'm pointing at addresses" (or the "I'm doing math… something something Dijkstra") people.
there's definitely something deeper and more significant between start at zero and start at one.
something which mathematics alone is not sufficient to uncover, it requires more foundations/logic to be relevant; it requires computer science (but focused as 'computology' and pure theory).
in mathematics this becomes more aparent with ring algebraic-structures; and when considering exponentials/logarithms/roots and powers.
a way to bring into focus what I'm trying to point out, is to consider the historical development of counting numbers. In a sense, it is not until the foundations are thoroughly considered (which is more logic than mathematics) that the importance of starting from zero really shines. In this 'historical sense' we started at one, and then went back and realized that it's more 'technically correct' to start at 0; this same foundational redefinition makes it important to note that zero is the only natural without a predecessor.
an alternative way to point at this 'deep and significant' issue, is to consider how zero is the additive identity and one is the multiplicative identity; algebraic-rings having both, and the distributive property being the axiom which links both. (I tend to focus on how both elements are identities; they're both instances of an 'identity' element; two sorts of identity).
Lol, well that's a terrible point to make. When a baby is born, it's 0 zero years old. Human beings literally start at zero...