Maximum matching M with the condition G[M] is 2K_2-free
Surprise! (for me).
This type of matchings are already studied in the literature. They are called connected matchings.

They were introduced by Plummer, Stiebitz and Toft in their study on Hadwiger conjecture. See the chapter "Connected Matchings" by Cameron in the book "Combinatorial Optimization – Eureka, You Shrink!"

The status of connected matchings in bipartite graphs (not necessary balanced) is open to the best of my knowledge (i shall update). The weighted version of the problem is NP-complete for bipartite graphs. The problem is polynomial time solvable for chordal bipartite graphs.

Update: the problem is NP-complete for balanced bipartite graphs (i.e., the exact problem asked in the question). This is proved in the paper "Multitasking Capacity: Hardness Results and Improved Constructions" by Alon et al.  They also report that finding the size of a largest connected matching is hard to approximate within a factor of $n^{1-\epsilon}$ unless NP = co-RP.

Notes added earlier (kept for interested people):
"Connected matchings in chordal bipartite graphs" by Jobson et al. (doi:https://doi.org/10.1016/j.disopt.2014.06.003) and "Connected matchings in special families of graphs" by Caragianis (thesis) are two notable references.