Can we obtain a larger model of ZF or ZFC in this way? Suppose we have a model $U$ of ZF (or ZFC). Can we always obtain a larger model by taking any collection of elements of $U$ and add it as a set to the model, and then add other sets so that the axioms continue to hold?
 A: There are several ways that one might answer this question.
First, the method of forcing, which has been used to prove so many of the set-theoretic independence results, such as the independence of the continuum hypothesis and the axiom of choice, works much like you describe. For a given model $M$ of ZFC, we may for certain kinds of subsets of $M$, the $M$-generic sets $G$, form a larger model $M[G]$. The new model is constructed essentially algebraically from the old, every set in the forcing extension $M[G]$ is definable from elements of $M$ and the new ideal object $G$.
Second, if one is concerned with well-founded models, or more specifically transitive models, which means models of set theory $M$, such that the membership relation of $M$ is the same as the ambient set-membership relation $\in$, then in this case, your principle follows from suitable large cardinal hypotheses. For example, if there are arbitrarily large inaccessible cardinals (this is equivalent to the Universe axiom often considered in category theory), then every set is an element of some $V_\kappa$ for inaccessible $\kappa$, which is a transitive set model of ZFC, and because these are as large as you like, you can find one containing your desired set. For your property, however, you don't actually need any inaccessible cardinals at all, for even worldly cardinals would suffice, a weaker notion. And indeed, there is no need for it to be $V_\kappa$s. Your principle (for transitive sets) is a consequence of the natural assumption that every set is an element of a transitive model of ZFC. That principle strictly exceeds ZFC in consistency strength, but is very mild by large cardinal standards, weaker even than a single inaccessible cardinal.
Finally, third, I think a fairly robust answer to your question is affirmative, by the following argument.
Theorem. For any model $M$ of ZFC, not necessarily well-founded, and any set $A\subseteq M$, there is a larger model $\bar M$ of ZFC, an elementary extension of $M$ (with all the same truths for objects in $M$), and with an object $a\in\bar M$, such that for every $x\in M$, we have $\bar M\models x\in a$ if and only if $x\in A$.
Proof. Let $T$ be the elementary diagram of $M$, which is all the truths of $M$ asserted in the language of set theory together with constants for every element $u\in M$, together with the theory, in a new constant symbol $a$, asserting for each $u\in M$ separately, either that $u\in a$, if it happens that $u\in A$, and asserting $u\notin a$, if it happens that $u\notin A$. Thus, $T$ expresses the theory that we want to be true in the model $\bar M$.
The main point now is that this theory $T$ is finitely consistent, since any finitely many assertions can be realized by interpreting the new constant symbol $a$ suitably in $M$. So by compactness, the whole theory is consistent. If $\bar M$ is a model of $T$, then we may identify $M$ with the interpretation of the constants $u$ in $\bar M$, and so we have the desired model. $\Box$
In this theorem, it might be important to notice that the object $a$ can have other elements from $\bar M$, which are not in $M$, and in this sense it is wrong to think of $a$ as representing exactly the set $A$. This feature is required, because for some models $M$, there are sets $A\subseteq M$ that simply cannot be represented exactly in any elementary extension. For example, if $M$ is ill-founded, then we could take a set $A$ revealing this, and this set cannot be added by itself, since $\bar M$ would see the ill-foundedness. But the theorem shows that we can find $\bar M$ with an object $a$ whose trace on $M$ is exactly $A$, even though $a$ gets new elements of $\bar M$ not in $M$.
Lastly, let me point out as mentioned by Asaf in the comments below, the general fact is that when building larger models of set theory, you often have to add quite a lot of new stuff, including perhaps new ordinals, new real numbers, new natural numbers even. In this sense, these extensions perhaps depart from the way we think of algebraic extensions of rings or fields.
