One of the main lessons of set theory is that many of our familiar set-theoretic concepts, such as countability, uncountability, well-orderedness, ill-foundedness, even finiteness, depend on the set-theoretic context in which they are considered. We know that different models of set theory can disagree about whether a set, the same set, is countable or uncountable, about whether a relation is well-founded or ill-founded and even about whether a set is finite or infinite.
First, of course, one can play all kinds of games using the compactness theorem or ultrapowers to make models with vastly different beliefs about the same set. For example, by starting in a universe $V$ and constructing a nonprincipal ultrapower $W=V^\omega/\mu$, one has nonstandard finite sets of natural numbers in $W$ that $W$ thinks are finite, but $V$ thinks have size continuum.
But second, the method of forcing allows us to construct new universes that stay close to the original universe, the ground model, in some ways while differing in others, which we can precisely control. They have the same ordinals, for example, and thus agree on the well-foundedness of the relations they have in common. But meanwhile, we can still make them disagree on other matters, such as the power set operation, since a forcing extension can have additional set of natural numbers and thus additional reals. So the concept of the "set of reals" is not absolute between models of set theory, even when they agree on the ordinals.
The issue of background dependence is sometimes thought not to apply to arithmetic, since we can prove that there is only one standard model of arithmetic. But one can object to this slight-of-hand: the uniqueness of $\mathbb{N}$ is proved in set theory, and so each model of set theory has a unique concept of natural number, but different models of set theory can have different non-isomorphic versions of $\mathbb{N}$, as my ultrapower example above shows.
Because of this pervasive dependency of set-theoretic concepts on the set-theoretic background in which they are considered, some set theorists, perhaps like your friend, are pushed to the view that every model of set theory provides its own concept of standardness, and that the notion of an absolute set-theoretic background is illusory. Every model of set theory look upon itself as the entire universe, a standard world by its own lights. This is a sense in which the notion of standardness itself is relative-to-a-model.
Other set-theorists retain the idea that there is an absolute sense of standardness, that we are living in the real universe of sets, and that it makes sense to judge models as being correct or incorrect about their judgements of well-foundedness. As you know, such questions were precisely the topic of my recent course on the philosophy of set theory, and the debate between these two perspectives is the thrust of my article on the set-theoretic multiverse, along with the other readings we had in the course.
But let me close by mentioning one case often mentioned by set-theorists, where a model of set theory can begin to have an inkling that something is seriously awry with its notions of standardness. Specifically, if ZFC is consistent, then we know that $\text{ZFC}+\neg\text{Con}(\text{ZFC})$ is also consistent. What would it be like to live in a model of this latter theory? In that universe, we would see that all the ZFC axioms are true, but also we would think that the ZFC axioms are formally inconsistent, admitting a proof of a contradiction.
