I’m not sure what you mean by “theorems remain the same”. If you take away induction from Peano arithmetic, you get Robinson arithmetic, which has many more models, including (from https://math.stackexchange.com/a/4076545):

- ℕ ∪ {∞}

- Cardinal arithmetic

- ℤ[x]⁺

Obviously, not all theorems that are true for the natural numbers are true for cardinals, so it seems misleading to say that theorems remain the same. I also believe that the addition of induction increases the consistency strength of the theory, so it’s not “just” a matter of expressing the theorems in a different way.

I would agree more for axioms that don’t affect consistency strength, like foundation or choice (over the rest of the ZF axioms).