How am I able to get the value out of cookie array when I push a button?

You need to put the id of each item inside form to indicate the item that will be deleted.

<?php 
$all_favorites= unserialize($_COOKIE['favorites']);

echo '<table>';
foreach($all_favorites as $key => $value) {
 echo '<tr>';
  echo 'Post-ID = ' . $value . ' ';
  ?>
  <form method="POST">
    <input type="hidden" name="id" value="<?php echo $value; ?>">
    <button type="submit" class="btn btn-default" name="delete">
        <span class="glyphicon glyphicon-trash"></span>
    </button>
  </form><br>
  <?php
 echo '</tr>';
}
echo '</table>';
if (isset($_POST['delete'])){
   $id = $_POST['id']; // do security checks (sanitize etc)
   // unset post with $id from cookie

}
?>