close InfoWindow before open another

The suggestion is only create one infowindow (in the global scope), reuse it and change its contents when a marker is clicked, close it if the user clicks on the map.

code snippet (removes the dependency on the AJAX call):

// global variables
var map;
var markers = [];
var infowindow = new google.maps.InfoWindow();

function initialize() {
  var mapOptions = {
    center: new google.maps.LatLng(44.49423583832911, 11.346244544982937),
    zoom: 13,
    mapTypeId: google.maps.MapTypeId.ROADMAP
  };
  map = new google.maps.Map(document.getElementById("mappa_locali"), mapOptions);
  var json = JSON.parse(data);
  for (var i = 0; i < json.length; i++) {
    point = new google.maps.LatLng(json[i].latitudine, json[i].longitudine);
    var infowindowHtml = '<a href="./dettaglioLocale.php?id_loc=' + json[i].id_locale + '">' + json[i].nome_locale + '</a><br>' + json[i].address;
    addMarkerz(point, infowindowHtml);
  }
}

function addMarkerz(point, infowindowHtml) {
  var marker = new google.maps.Marker({
    position: point,
    map: map
  });
  google.maps.event.addListener(marker, 'mouseover', infoCallback(infowindowHtml, marker));
  markers.push(marker);
}

function infoCallback(infowindowHtml, marker) {
  return function() {
    infowindow.close();
    // update the content of the infowindow before opening it
    infowindow.setContent(infowindowHtml)
    infowindow.open(map, marker);

  };
}


var data = JSON.stringify([{
  latitudine: 44.494887,
  longitudine: 11.3426163,
  id_locale: "0",
  nome_locale: "Bologna",
  address: "Bologna, Italy"
}, {
  latitudine: 44.4946615,
  longitudine: 11.314634999999953,
  id_locale: "0",
  nome_locale: "Quartiere Saragozza",
  address: "Quartiere Saragozza, Bologna, Italy"
}]);


google.maps.event.addDomListener(window, "load", initialize);

html,
body,
#mappa_locali {
  height: 100%;
  width: 100%;
  margin: 0px;
  padding: 0px
}

<script src="https://maps.googleapis.com/maps/api/js"></script>
<div id="mappa_locali"></div>

function infoCallback(infowindow, marker) { 
  return function() {
     infowindow.close();  
    infowindow.open(map, marker);

  };
}

should be changed to

function infoCallback(infowindow, marker) { 
    return function() {
        //Close active window if exists
        if (activeWindow != null) {
            activeWindow.close();
        }
        //Open new window 
        infowindow.open(map,marker); 
        //Store new window in global variable 
        activeWindow = infowindow; 
    };
}

and declare activeWindow as a global variable in your .js file as var activeWindow.

I think I have a simple solution. I just save the last open mark. On the next click first of all I close the last mark.

var markers = [];
var infowindows = [];

function addMarkes(facilityID) {

        var bounds = new google.maps.LatLngBounds();

        for (var i = 0; i < variants.length; i++) {

            var v = variants[i];

            var position = new google.maps.LatLng(v.Lat, v.Long);

            bounds.extend(position);

            markers[i] = new google.maps.Marker({
                position: position,
                title: v.Title,
                map: map,
                animation: google.maps.Animation.DROP,
                icon: v.Icon,
                infoWindowIndex: i //synchronize with infoWindows
            });

            var localContent = '<div><span class="address">' + v.Address + '</span></div>';

            infowindows[i] = new google.maps.InfoWindow({
                content: localContent
            });

            //Just for multiple marks. 
            if (variants.length > 0) {
                var lastOpen = -1; //Save the last open mark
                google.maps.event.addListener(markers[i], 'click', function (innerKey) {
                    return function () {
                        //Close the last mark.
                        if (lastOpen > -1) {
                            infowindows[lastOpen].close();
                        }
                        infowindows[innerKey].open(map, markers[innerKey]);
                        lastOpen = innerKey;
                    }
                }(i));
            }
        }

        map.fitBounds(bounds);

    }