In ZF, the Axiom of power set implies that for any set, there exists a set that contains all sets known to be subsets of such. However, as noted by the answer to this question:
It doesn't guarantee that any particular subsets exist, only that whenever you find something in your model that is a subset of X, it will be in P(X).
What axioms would be required to imply that any particular subsets exist? I.e; given any (finite or infinite) selection of elements of some set, there exists a subset with (or without) exactly that selection?
One could simply demand that every subset of any set in a given model is a set in that model. In slightly more explicit terms, if $x\in M,$ and $Y\subseteq M$ is any collection of elements of $x$, then there is a $y\in M$ such that for any $z\in M,$ $z\in Y\iff zEy.$ (Where $E$ is the membership relation for $M$.) I think I've heard this property called fullness, or swelled-ness (let's stick with fullness).
However, this is not an axiom in the usual sense, because it cannot be expressed as a first-order statement about the model. It is a second-order condition, since it quantifies over the $Y$, which are classes, not sets in $M.$ And we know from basic results in model theory that there is no way that we can make this first order, since first-order conditions do not even allow us to control the cardinality of the model. For instance, if there are models of ZF (or any other set theory), then there are countable models, in which there are necessarily only countably many subsets of $\mathbb N,$ so we know such a model cannot be full in the above sense.
That is the "external" perspective, anyway, where we are talking about a model of set theory from the outside. From the inside of a model, or equivalently when we are just working in ZFC and talking about the "real" universe of sets, there's no notion of having "all" of the subsets of a given set. Or, rather, it's a trivial notion: of course we have all of the subsets, that is what the word "all" means. There are certain sets that we can define (e.g. the set of even numbers) but there could be others that we can't (e.g. some random subset of $\mathbb N$ with no discernible properties whatsoever), and there's not really even a sharp notion of 'making sure we have all of those'.
Which is how we can have all these different models with very different collections of subsets of a given defined set in the first place. (And also sometimes very different answers to fundamental questions like the continuum hypothesis).