This is possible but you need the save the posted data from the contact form to the session and show it there.
Add this to your functions.php
add_action('wpcf7_mail_sent', 'save_cf7_data');
function save_cf7_data($cf)
{
if(session_id() == '') {
session_start();
}
$current_submission = WPCF7_Submission::get_instance();
$_SESSION['cf7_submission'] = $current_submission->get_posted_data();
}
And your success page you just need to print the session var, like:
echo $_SESSION['cf7_submission']['name'];
That's all.