How is $\mathbb N$ actually defined? I know perfectly well the Peano axioms, but if they were sufficient for defining $\mathbb N$, there would be no controversy whether $0$ is a member of $\mathbb N$ or not because $\mathbb N$ is isomorphic to a host of sub- and supersets of $\mathbb Z$.
So it seems that mathematicians, when specifying $\mathbb N$, rarely mean the set of natural numbers according to its definition but rather some particular embedding into sets defined by different axiomatic systems.
Doesn't this sort of misstate then the role of the Peano axioms as the defining characteristics of $\mathbb N$?  It seems like most of the time the Peano axioms are not more than a retrofit to a subset of $\mathbb Z$ or other sets defined by a different axiomatic system with one or several axioms incompatible with the Peano axioms.

You have to start somewhere.
Some treatments of mathematics start with Peano's axioms for the natural numbers, and use that to define other number systems which extend the natural numbers --- first integers, then rational numbers, real numbers, complex numbers,...
But that's not the only choice. You could equally well decide to start with the axioms for the real numbers, and use that to define other number systems: first natural numbers, integers, and rationals within the real numbers, then the extension to complex numbers …
The above two methods can be thought of as dependent on "naive set theory". Because of that, perhaps a more theoretically satisfying place to start is with axioms for set theory such as ZF (the Zermelo Frankel axioms) or ZFC (the Zermelo Frankel axioms with the axiom of Choice tacked on). In these treatments, the natural numbers are usually defined as the set of finite ordinals (the existence of which is one of the ZF axioms, and the Peano axioms must then be proved), in which case the number zero, which is defined as the empty set, would be included. Alternatively, the natural numbers could be defined as the finite nonempty ordinals, in which case zero would not be included.
As you can see, there's different choices of where to start, and different people may make different choices. 
