There must be a clog in your drain for the water to build up enough to come out that hole. I just found 2 similar holes at the back of my vessel sink yesterday when I plunged the sink, and the water all came out those holes onto the floor. What I know so far is that the overflow hole leads to a cavity between the top and bottom of the sink, where water feeds into openings in the drain pipe, and then goes down the drain. In my case, the holes at the back of my sink connect to that cavity. So when I plunged, the water went out the holes in the drain stem, into the cavity, and out the holes in the back of the sink. I took the sink out, and there is no significant clog in the overflow or through the p-trap, so today I will check for a clog beyond the wall. The sink has been draining slowly, which is why I plunged. Also, in taking the sink apart, I found black gunk on the stopper and pipes. This is bacteria that gathers on soap scum, etc, probably another symptom of a clog, and should be cleaned. Be careful if you plunge, because the water might all come out that hole.