Step 1
Be sure you have moment.js and moment.js locale correctly in your head script section, i'm using cdn:
<script src="https://cdnjs.cloudflare.com/ajax/libs/moment.js/2.29.2/moment.min.js"></script>
              
<script src="https://cdnjs.cloudflare.com/ajax/libs/moment.js/2.29.4/locale/fr.min.js"></script>
Step 2
Change your column render function to this:
render: function(data, type, full) {
                  return    moment(new Date(data)).locale('en-es').format('MMMM') + " | " +
                            moment(new Date(data)).locale('fr-fr').format('MMMM');
                }
Output

Full code
<html>
    <head>
        <script
              src="https://code.jquery.com/jquery-3.6.0.min.js"
              integrity="sha256-/xUj+3OJU5yExlq6GSYGSHk7tPXikynS7ogEvDej/m4="
              crossorigin="anonymous"></script>
              
        <script
              src="https://cdn.datatables.net/1.12.1/js/jquery.dataTables.min.js"></script>
              
        <link rel="stylesheet" href="https://cdn.datatables.net/1.12.1/css/jquery.dataTables.min.css">
        
        <script
              src="https://cdnjs.cloudflare.com/ajax/libs/moment.js/2.29.2/moment.min.js"></script>
              
        <script
              src="https://cdnjs.cloudflare.com/ajax/libs/moment.js/2.29.4/locale/fr.min.js"></script>
    </head>
    <body>
    
        <table id="example" class="display" style="width:100%">
            <thead>
                <tr>
                    <th>Name</th>
                    <th>Position</th>
                    <th>Office</th>
                    <th>Age</th>
                    <th>Start date</th>
                    <th>Salary</th>
                </tr>
            </thead>
            <tbody>
                <tr>
                    <td>Tiger Nixon</td>
                    <td>System Architect</td>
                    <td>Edinburgh</td>
                    <td>61</td>
                    <td>2011-04-25</td>
                    <td>$320,800</td>
                </tr>
                <tr>
                    <td>Garrett Winters</td>
                    <td>Accountant</td>
                    <td>Tokyo</td>
                    <td>63</td>
                    <td>2011-07-25</td>
                    <td>$170,750</td>
                </tr>
                <tr>
                    <td>Ashton Cox</td>
                    <td>Junior Technical Author</td>
                    <td>San Francisco</td>
                    <td>66</td>
                    <td>2009-01-12</td>
                    <td>$86,000</td>
                </tr>
            </tbody>
            <tfoot>
                <tr>
                    <th>Name</th>
                    <th>Position</th>
                    <th>Office</th>
                    <th>Age</th>
                    <th>Start date</th>
                    <th>Salary</th>
                </tr>
            </tfoot>
        </table>
    </body>
</html>
<script>
$(document).ready(function () {
    $('#example').DataTable( {
        "language": {
            url: 'https://cdn.datatables.net/plug-ins/1.11.5/i18n/fr-FR.json',
        },columnDefs: [
            {
                targets: 4,
                render: function(data, type, full) {
                  return    moment(new Date(data)).locale('en-es').format('MMMM') + " | " +
                            moment(new Date(data)).locale('fr-fr').format('MMMM');
                }   
            }
        ]
    } );
});
</script>