Right, it proves S(2)->S(3), but induction asks you to prove S(n)->S(n+1) for n in general, and not just for some n. The inductive proof doesn't work for S(1)->S(2), so it clearly can't work for the more general n->n+1.
However, if it were possible to prove the case n = 2, we would have a valid inductive proof for n >= 2.
Right, so it doesn't work -- either it's a correct proof of a non-sequitur ("true for n implies true for n+1, provided n meets some criteria"), or an incorrect proof of an inductive step ("true for n implies true for n+1").
Because it's claimed to be proof by induction, it's meant to be the latter -- the person doing the proving claimed to have proven the inductive step, and their proof of it was incorrect.
However, it must be coupled with a base case >= 2, which isn’t the case here. The only base case proven is 1.
See also “Induction basis other than 0 or 1” [0].
[0] https://en.wikipedia.org/wiki/Mathematical_induction#Inducti...
Nobody was trying or claiming to be doing any other kind of induction. They said they had the base case and the inductive step, and they were clear about the base case and the inductive step. Their proof of the base case was correct, and their proof of the inductive step did not prove what they said it did. It doesn't matter what it did prove.