Mouthwash is typically alcohol based. Alcohol is a pretty good disinfectant in general.
But it's efficacy will really only be decent while it's in your mouth. Once it gets diluted past a certain point, it's not going to be doing anything. You'd probably have similar results with vodka.
I know doctors who recommended drinking whiskey early in the pandemic for that reason (and also the usual reasons people dealing with trauma reach for whiskey)
I was actually surprised by the mouthwash outcomes as well. Almost no one really talked about mouthwash, but it looked to be useful in the study.