How to send and process N distinct forms with ajax without giving refresh on the page?

First, you can not have multiple elements with the same id , as commented above @Andrey. Correct this in your HTML or it will be invalid (out of the problems you will have when trying to locate these repeated elements by id). You can replace the ids with classes.

Simply submitting all forms to Ajax, using the serialize method of jQuery, which scans a form and collects the values from the fields. The example below uses Promises to detect when everything is complete.

var promessas = [];
var url = "url_do_seu_php";
$("form").each(function(){
  var dados = $(this).serialize();
  promessas.push($.post(url, dados));
});
$.when.apply($, promessas).done(function(){
  // todos os formulários foram processados
});