Maybe a roundabout answer to your question, but Peano's axioms are equiconsistent with many finite set theories (even ZFC without axiom of infinity), and I do think philosophically it makes more sense to say weak axiomatic set theory + predicate calculus forms building blocks of arithmetic[1]. The idea of "number" as conceived by Frege is an equivalence class on finite sets: A ~ B <-> there is a bijection, which is in fact a good way of explaining "counting with fingers" as an especially primitive building block of arithmetic:

  {index, middle, ring} ~ 
  {apple, other apple, other other apple} ~
  {1, 2, 3}

as representatives of the class "3" etc etc, predicates would be "don't include overripe apples when you count" etc. Then additions are unions and so on, and the Peano axioms are a consequence.

[1] In my view Peano axioms are the Platonic ideal of arithmetic, after the cruft of bijections and whatnot are tossed away. I agree this is splitting hairs.

Another one is Presburger Arithmetic, which is Peano Arithmetic minus the multiplication. What makes it interesting (and useful) is that this removal makes the theory decidable.

https://en.wikipedia.org/wiki/Presburger_arithmetic

Conway's Surreal Numbers https://en.wikipedia.org/wiki/Surreal_number#Arithmetic

There are many axiom systems for natural numbers. My favorites are 1. Heyting Arithmetic, and 2. In category theory, one can characterise the natural numbers and the initial algebra of the X |–> X+1. functor.