Imagine for example the following structure. It consists of the natural numbers, coloured blue, together with the integers, coloured red. The successor operation is the natural one.
If you want a more formal description, our structure $S$ is the union of the set of all ordered pairs $(a,0)$, where $a$ ranges over the natural numbers, and the set of all ordered pairs $(b,1)$, where $b$ ranges over the integers. If $x=(a,0)$, define $S(x)$ by $S(x)=(a+1,0)$, and if $x=(b,1)$, define $S(x)$ by $S(x)=(b+1,1)$.
This structure $S$ satisfies your axiom, but is quite different from the natural numbers.
There are much worse possibilities. In the above description of $S$, instead of using all pairs $(b,1)$ where $b$ ranges over the integers, we can use all pairs $(b,1)$, where $b$ ranges over the reals.
Remark: Because of the wording of the question, we addressed only the issue of order type, which is settled by the second-order version of the induction axiom, and is indeed equivalent to it. Order types of models provide only weak information about the structure of models of first-order Peano arithmetic.
Use induction on the statement "$n=0$ or there exists a natural number $b$ such that $b++=n$."
Showing this for $n=0$ is obvious. Do you need help on showing the induction step?
ADDED: Here is the induction step (in summary: you can fill in the small gaps).
Assume the statement is true for $n$. There are two cases: $n=0$ or $n \ne 0$.
If $n=0$, then $0++=n++$, and the statement is true for $n++$.
If $n\ne 0$, we must have $b++=n$ for some $b$ (since we assumed the statement is true for $n$). Then $(b++)++=n++$, and again the statement is true for $n++$.
ADDED EVEN LATER:
@Andreas Blass gave a much shorter and better induction step in his comment. Here it is:
Assume the statement is true for $n$ (although we will not use this assumption). Then let $b=n$. We trivially get $b++=n++$, so the statement is true for $n++$.
Best Answer
I think you are missing the point of the Peano Axioms. It postulates that there is a set $\,\mathbb{N}\,$ which is by convention called the set of natural numbers. We are given that zero is a natural number and is by convention denoted by $\,0.\,$ In order to avoid confusion and emphasize its nature, perhaps it would be better to use a distinct notation such as "$0$". The successor of zero is a natural number by convention denoted by "$1$". This conventional defining property of "$1$" as the successor of "$0$" in conjunction with other definitions and the Peano Axioms leads to all of its properties. Similarly for all of the other natural numbers. Each natural number is defined to be the successor the the previous number. In other words, Only the number zero is initially given and all of the rest of the natural numbers are determined as the repeated successors of zero. For example,
$$ \text{"}1\text{"} :=S(\text{"}0\text{"}),\; \text{"}2\text{"}:=S(S(\text{"}0\text{"})),\,\dots .\,$$
The actual identity of the other natural numbers is not important. What is of great importance is that they are all definite successors of zero. You are allowed to use any given set as the set of natural numbers as long as one element is singled out as the zero element and all the rest of the elements are the successors of zero.
Thus, if you wish, you can use the set of even numbers as a model of the natural numbers and then define $\,2=S(0),\,$ $4=S(2),\,$ $6=S(4),\,$ $\dots.\,$ In this model the number denoted by $\,2\,$ is the successor of zero but this does not change its properties in the model of the natural numbers. For example, in this model, we have $\,2\times 2=2\,$ using the Peano definition of $\,\times\,$ (multiplication) of natural numbers. This is because $\,2\,$ models the natural number "$1$" and has the same properties in the model as "$1$" has.
This is an example of the abstract nature of modern axiomatic mathematics. The natural numbers are not defined by what they are, but by what they do. All models of the natural numbers based on the Peano Axioms are equivalent in the sense that they all have the same properties in the model. The Wikipedia article Peano axioms has a lot of details, but the fundamental idea is that it is an axiomatic model of the natural numbers. Previously generations of mathematicians took them as given somewhat as in the quote "God made the integers, all else is the work of man". The natural numbers have not changed, but our view of their nature has.