If control units are matched exactly to treated units such that Xi = Xj then we can say that this is the estimated Average Treatment Effect (ATE) where ATE = E [Y(1) - Y(0)].
Fair enough, but this does ignore the fact the Controls ... were not treated!
It's seldom addressed that for all the effort to match on observed covariates, there is some reason somewhere why the Case was treated and the Control was not.
Rosenbaum (elsewhere) says this can be collapsed to one unmeasured covariate but practically, there'll be a host of reasons the why the Control was 'wrongly' not treated (or indeed, the Case was 'wrongly' treated),