Consider a countable set like $\mathbb{Q}$. Let $f : \mathbb{N} \rightarrow \mathbb{Q}$ be any bijection witnessing the equinumerosity of $\mathbb{Q}$ And $\mathbb{N}$. Then for each predicate $\phi$, the induction principle for $\mathbb{N}$ implies that $(\forall p \in \mathbb{Q})\phi(p)$ whenever $\phi(f(0))$ and $(\forall n \in \mathbb{N})(\phi(f(n)) \rightarrow \phi(f(n + 1)))$. More generally, each such $f$ induces a well-ordering on the relevant set, from which an induction principle for that set can be derived.
Is there an interesting example of a property true of some set like $\mathbb{Q}$ or $\mathbb{Z}$, the proof of which proceeds by induction in the manner described? It's quite easy to construct trivial examples to illustrate the possibility. However, I've never seen and can't seem to find any real examples.
Essentially you're using a wellordering of $\mathbb{Q}$ (of length $\omega$) and proving the statement by induction on the wellordering. In this particular case we can actually construct, effectively specify, the wellordering.
For general infinite sets, we can't, and we have to appeal to the Axiom of Choice (AC). Sierpinski wrote a monograph Hypothese du Continu (1934) (The Continuum Hypothesis, CH), which explores consequences of CH. If I recall correctly, some proofs and/or constructions therein proceed from a wellordering of the reals of length $\omega_1$, guaranteed by AC + CH. Such a wellordering has the nice property that every initial segment is countable.
Though I can't recall further examples right now, I have seen early 20th century proofs which use a wellordering of the set under consideration, where in more recent times one would use a different form of AC — Zorn's lemma or the like.