What is wrong with this induction proof? I am confused on how this is wrong other than the base case being wrong.

What makes the proof wrong is that, when $n=0$, you cannot multiply by $(n+1)/n$,

because that would involve dividing by $0$, so it is not allowed.

For a different kind of induction fallacy, see this Wikipedia article.

You have divided by $n$ in your induction step where you have multiplied by $(n+1)/n$

Note that you may not divide by $n=0$ but your initial step was involving $n=0$

Thus your argument fails at $$P(0)\implies P(1)$$