When learning Peano axioms normally we use induction to suggest that if a property $P(0)$ is true, and if $P(k)$ being true implies $P(k + 1)$ being true (for natural number $k$), then if $P$ is the property of being a natural number, we start with $0$ being a natural number by definition, then by induction $1$ is a natural number, $2$ is a natural number, $3$ is a natural number, etc.
This is done to prohibit things like $5.3$ or $\pi$ from being natural since they are not reachable when applying the successor function from $0$.
But what I don't understand:
Just because we can point to the $0, 1, 2, 3, ...$ chain as being natural numbers, what stops us from making some other chain and calling them natural numbers?
If your answer to #1 is "Well we never provided a mechanism for saying that they were natural numbers to begin with" then what was the "risk" or issue if they posed no threat? Why do we have to specifically label the stuff connected to $0$ as natural numbers when it's more or less implied that the rules we're setting up are specifically for defining what rules natural numbers follow?