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 $\,0.\,$ The successor of zero is a natural
number is by convention denoted by $\,1.\,$ According to
Peano postulates the defining property of $\,1\,$ is that
it is the successor of $\,0.\,$ 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, $\,1:=S(0),\,2:=S(S(0)),
\,\dots.\,$ The actual identity of the other natural numbers
is not important. What is important is that they are the
successors of zero. You are allowed to use any 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 Peano natural numbers 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 mathematicians took them as given somewhat
as in the quote "God made the integers, all else is the work of man".