$wpbd->insert() does not insert user data

user_register action allows you to access data for a new user immediately after they are added to the database. The user id is passed to hook as an argument.

So you wish to insert record for user when user register you can modify your code like follows:

add_action( 'user_register', 'nuevoPostulante', 10, 1 );
function nuevoPostulante( $user_id ) {
    global $wpdb;
    $tablaPostulante = $wpdb->prefix . 'postulante';
    $current_user = get_user_by( 'ID', $user_id ); // You can use get_userdata( $user_id ) instead of get_user_by() both can be work perfectly suite your requirement.
    $wpdb->insert(
        $tablaPostulante,
        array(
            'dni' => $current_user->user_login,
            'nombre' => $current_user->display_name,
            'email' => '[email protected]',
        )
    );
}