The main lesson is that it doesn't matter at all which particular objects you take as the numbers and what function you use as the successor function, as long as your system fulfills the right structural features.
It was Dedekind who first identified this structural theory in 1888 and proved his famous categoricity theorem, which showed that there is up to isomorphism only one model of his theory. His axioms were as follows:
Dedekind arithmetic. For a set $N$ of objects that we will call the natural numbers, with a distinguished element $0$ in $N$ called zero, and a unary function $S:N\to N$ called the successor function, the following three axioms:
- zero is not a successor, that is, $0\neq Sx$ for all $x\in N$
- the successor function is one-to-one, that is, $Sx=Sy\to x=y$, and
- the successor function generates every number from zero, in the sense that if $X\subseteq N$ is a set for which $0\in X$ and $x\in X$ implies $Sx\in X$, then every number is in $X$, that is, $X=N$.
Notice that the axioms do not refer to any essential properties of the elements of $N$ or the function $S$---they can be anything at all, as long as the system fulfills the stated structural properties.
But furthermore, as emphasized in the comments by KP Hart, one doesn't really have the natural numbers and the operation $x+1$ and so forth until after introducing Dedekind's theory and then developing number theory on the basis of it, defining $x+y$ and $x\cdot y$ and $x^y$ by recursion, as he did and as Peano did so successfully after him, using his theory.
Theorem. (Dedekind 1888) Any two models of Dedekind arithmetic are isomorphic.
If you have your natural number structure $\langle\mathbb{N},0,S\rangle$ and I have my natural number structure $\langle \bar{\mathbb{N}},\bar 0,\bar S\rangle$, then there will be an isomorphism $\pi:\mathbb{N}\to\bar{\mathbb{N}}$, a bijection of the sets such that $\pi(0)=\bar 0$ and $\pi(Sx)=\bar S(\pi(x))$ for every $x\in\mathbb{N}$.
Indeed, the requirement of the isomorphism tells you exactly how to define $\pi$, namely, by recursion, setting $\pi(0)=\bar 0$ and $\pi(Sx)=\bar S(\pi(x))$. So, in order to prove his theorem, Dedekind first proved that definition by recursion is legitimate in any model of his theory. And then the categoricity result drops out of it.
In my view, Dedekind's theorem is the beginning of the philosophy of structuralism in mathematics, the view that we in mathematics we should treat all our mathematical ideas and structure as invariant under isomorphism. We should not seek to find out what numbers "really" are, to find their essence, but rather to find the structural features that characterize the numbers as we intend them. This is just what Dedekind did.
We now have categorical accounts for all of our familiar structures---the reals are up to isomorphism the unique complete ordered field (Huntington 1904); the complex numbers are up to isomorphism the unique algebraic closure of the real field; the various ranks of the set-theoretic hierarchy also have categorical characterizations (Zermelo 1930). And so on. These categorical characterizations show us how to do mathematics in an isomorphism invariant manner.
There is nothing mathematical at stake in having a different copy of the natural number structure.
(I have noticed that some people are downvoting your question, because perhaps it isn't asked in the best way. But I find the topic both philosophically interesting and deep, and so I have posted this answer. Perhaps the question will get migrated to math.stackexchange.)
