Let $G$ be a (possibly infinite) simple graph and assume there is a series of matchings (i.e. independent sets of edges) $E_1,E_2,E_3,\ldots$ such that their cardinalities fulfill $|E_1|\subseteq|E_2|\subseteq|E_3|\subseteq\ldots$, is there then a matching $E$ such that $$|E|=\sup_{n\in\mathbb{N}}|E_n|\;?$$ This is trivially true if $\{|E_n|\}_{n\in\mathbb N}$ becomes eventually constant. I think it is also true for $|E|=\omega$, the argument being for a given matching $E'_m$ with $|E'_m|=m$ there surely is some $E_N$ with $N>m$ such that $E_N$ contains an edge that is independent of $E'_m$, so we can take $$E=\bigcup_{m\in\mathbb N} E_m.$$ The "surely" of the last sentence is the weak point of course. My thought is that in a finite-branching graph the edge set incident with $E'_m$ must be finite, so eventually there is an $E_N$ with an edge outside of it, and in the other case even if $E'_m$ incides with infinite star subgraphs of $G$ there must be enough edges left outside of these stars or $\{|E_n|\}_{n\in\mathbb N}$ would become eventually constant. The argument is not rigid, but I can imagine it could work.
However, I can't imagine a proof for $|E|$ being an arbitrary cardinal, since then the "take one extra edge per step" approach doesn't work anymore. In case the statement is also false for $|E|=\omega$ because I missed something, I would be interested if it still holds for bigger cardinalities.
I think that your argument is fundamentally sound and it could be extended to work with larger cardinalities by using transfinite recursion. Below I'll give a different formulation, but the idea is essentially the same.*
By a standard application of Zorn's lemma, $G$ has a maximal matching $M$ (and in fact every matching can be extended to a maximal matching). This implies that the set of vertices $V(M)$ of $M$ is a dominating set in $G$, i.e. every edge has at least one endpoint in $V(M)$. Then for any matching $M'$ of $G$ we have $|M'| \leq |V(M)|$: mapping each edge in $M'$ to an endpoint that is in $V(M)$ gives an injection from the former to the latter.
In particular, since $|V(M)| = 2|M|$, we have $|M'| \leq 2|M|$. It follows that if there is an infinite matching in $G$, then all the maximal matchings have the same (infinite) cardinality, and if $G$ has no infinite matchings, then the size of a maximal matching in $G$ is bounded. These observations imply your proposition for any cardinality.
Of course, this uses the axiom of choice in multiple places. I don't know whether this can be avoided, though my guess would be that the answer is no.
* I like the title of the following article: Zorn’s lemma is what happens when you get bored of transfinite induction.