 Solution

No. In the six-cycle with cyclic vertices $1,2,\dots ,6$, every edge belongs to one of the two alternating perfect matchings. However, the disjoint edges $12$ and $45$ cannot belong to a common perfect matching: the remaining vertices $3,6$ are not adjacent. Taking $X=\{1,3,5\}$ and $Y=\{2,4,6\}$ gives the required bipartite graph counterexample.