JSFiddle - React, Tailwind, and code Playground

HTML

<div id="wrapper">
    <div id="product_name">
        <input id="skriv_produktnavn" placeholder="Skriv Produktnavn her" required />
        <button id="add_another_product_name">Tilføj endnu et produktnavn</button>
        <div id="add_remove_product_name">
            <input id="added_product_name" placeholder="Skriv Produktnavn her" required />
            <button id="remove_product_name">X</button>
        </div>
    </div>
</div>
<button id="clone">Clone</button>

JavaScript

var $tpl = $('#product_name').clone();
var num = 0

$('#clone').click(function () {
    num++;
    var $cloned = $tpl.clone();
    $cloned.attr('id', $tpl.attr('id') + '_' + num);
    $(':not([id=""])', $cloned).each(function(){
        $(this).attr('id', $(this).attr('id') + '_'+num); 
    });
    $cloned.appendTo('#wrapper');
});