zoneNameField is used to build a query which is required to create a feature layer containing one polygon. You could use the objectID field for this value too.
queryString = '"' + zoneNameField + '" = ' + "'" + zoneName + "'"
arcpy.MakeFeatureLayer_management(Zones, "CurrentZoneLayer", queryString)
And yes, if you had multiple polygons with the same name, they would get grouped together for the counts because the query would select them all.