All of these models appear to contain infinitely sized objects that are explicitly named / manipulable within the model, which makes them extensions of the Peano numbers though, or else they add other, extra axioms to the Peano model.
If you (for example) extend Peano numbers with extra axioms that state things like “hey, here are some hyperreals” or “this Goedel sentence is explicitly defined to be true (or false)” it’s unsurprising that you can end up in some weird places.
We are able to recognize that they are nonstandard because they contain numbers that we recognize are infinite. But there is absolutely no statement that can be made from within the model from which it could be discovered that those numbers are infinite.
Furthermore, it is possible to construct nonstandard models such that every statement that is true in our model, remains true in that one, and ditto for every statement that is false. They really look identical to our model, except that we know from construction that they aren't. This fact is what makes the transfer principle work in nonstandard analysis, and the ultrapower construction shows how to do it.
(My snark about NSA is that we shouldn't need the axiom of choice to find the derivative of x^2. But I do find it an interesting approach to know about.)
No additional axioms are needed for the existence of these models. On the contrary additional axioms are needed in order to eliminate them, and even still no amount of axioms can eliminate all of these extensions without introducing an inconsistency.
If you (for example) extend Peano numbers with extra axioms that state things like “hey, here are some hyperreals” or “this Goedel sentence is explicitly defined to be true (or false)” it’s unsurprising that you can end up in some weird places.