create table file_multi (id int NOT NULL AUTO_INCREMENT, filename varchar(255), PRIMARY KEY(id));

<form action="file_upload.php" name="upload_form" method="POST" enctype="multipart/form-data">
    Файлы:<br />
    <input type="file" name="file[]" multiple />
    <input type="submit" value="Отправить" />
</form>

<?php
$host = "localhost";
$username = "user";
$password = "pass";
$dbname = "dbname";
$conn = new mysqli($host, $username, $password, $dbname);
if ($conn->connect_error) {
    die("Connection failed: " . $conn->connect_error);
}
foreach ($_FILES['file']['name'] as $filename) {
    $sql = "insert into file_multi values(NULL, "{$filename}")";
    $conn->query($sql);
}
$conn->close();

<?php
$dir = 'd:uploads\';//В linux в другую сторону слэши
$host = "host";
$username = "root";
$password = "pass";
$dbname = "dbname";
$conn = new mysqli($host, $username, $password, $dbname);
if ($conn->connect_error) {
    die("Connection failed: " . $conn->connect_error);
}
//Более корректный перебор по ключу и значению
foreach($_FILES['file']['name'] as $k=>$f) {
    if (!$_FILES['file']['error'][$k]) {
        if (is_uploaded_file($_FILES['file']['tmp_name'][$k])) {//Проверка загрузки файла во временное хранилище
            if (move_uploaded_file($_FILES['file']['tmp_name'][$k], $dir . $_FILES['file']['name'][$k])) {//Перемещение временного файла
                echo '<p>Файл: '.$_FILES['file']['name'][$k].' загружен.</p>';
            }
        }
    }
    $sql = "insert into file_multi values(NULL, "{$_FILES['file']['name'][$k]}")";
    $conn->query($sql);
}
echo "Успешно разружены все файлы";
$conn->close();