How does Peano Postulates construct Natural numbers only?

Yes, you can find other sets on which a successor function is defined that satisfies all the Peano axioms.

What makes the natural numbers unique is that you can use the Peano postulates to prove that when you have two such sets you can build a bijection between them that maps one successor function to the other. That means the sets are really "the same" - the elements just have different names.

So you might as well use the traditional names $ 1, 2, 3,\ldots$.

This axioms define the natural numbers up to isomorphism; That is, given another Peano system $S$, there always is a bijective function $f: \mathbb{N}\to S$ such that $f(s(n))=s_1(f(n)); f(1)=1_S$. This substantially means that from the Peano's structure point of view, the two are equal. Given $1, 1_S$, we can just identify as equal the two of them and the other numbers will follow, as if we called the natural numbers with different symbols $1_S$ instead that $1$, but the structure behind is the same (this theorem can actually be formally proved by using the principle of recursion)

A little note, though: this isomorphism is sometimes not very evident end extremely helpful. For example, $\mathbb{N}^2$ can be given a Peano structure, and this can be identified with $\mathbb{N}$ by means of an isomorhpism (namely Cantor pairing function), which is highly non trivial.