There is in fact a topological(?) proof of this statement and the following generalization, essentially due to Péter Csikvári (Section 4, Lemma 4.2).
We define a double cover (or ''2-lift'') $H$ of a graph $G$ as follows:
consider $G$ as a topological space with CW structure, and $\pi:H\to G$ is
a topological double cover with CW structure induced from $G$.
Let $G\sqcup G$ denote the disconnected double cover of $G$. Let $m_k(G)$ denote the number of $k$-edge matchings in $G$.
Prop. Let $G$ be a graph with no cycle of length smaller than $g$ (e.g. let $g$ be the girth of $G$) and let $H$ be a double cover of $G$.
Then for any $k < g$,
$$m_k(H) = m_k(G\sqcup G) .$$
Proof: Consider a $k$-matching $M\subset H$, and consider its image $\pi(M)$ in $G$. The image has vertices of valence at most 2, so $\pi(M)$ is a disjoint union of paths and cycles. But since $k<g$, there are no cycles. Finally, observe that any path in $G$ lifts to a matching in $H$ in exactly two ways, where $H$ is any double cover. Since $G\sqcup G$ is a double cover, the result follows.
Csikvári observed that when $G$ is bipartite, the same type of argument, now accounting for (even) cycles, implies that
$$m_k(H) \leq m_k(G\sqcup G) $$
for any size matching $k$.
This is used to prove tight lower bounds on the number of matchings in a bipartite, $d$-regular graph.