You are right. Numbers in Peano's axioms are variables that are open to all kinds of interpretations.
As Bertrand Russell pointed out, given any progression, say a row of pebbles, $p_0$, $p_1$, $p_2$, ..., $p_n$, ..., all of Peano's axioms will be true of this series.
And if we define:
> 1. R as the relation that relates $p$ to the next $p$,
> 2. $p_m+p_n$ as the $p$ to which the relation $R^n$ relates $p_m$,
> 3. $p_m \times p_n$ as the $p$ to which the relation $(\overset{\smile}{R}|R^n)^m$ relate $p_m$