Fluent Form Login by Username.php

add_action('fluentform_before_insert_submission', function ($insertData, $data, $form) {

if($form->id != 229) { // 229 is your form id. Change the 229 with your own login for ID
return;
}
$redirectUrl = home_url(); // You can change the redirect url after successful login

if (get_current_user_id()) { // user already registered
wp_send_json_success([
'result' => [
'redirectUrl' => $redirectUrl,
'message' => 'Your are already logged in. Redirecting now...'
]
]);
}

$username = \FluentForm\Framework\Helpers\ArrayHelper::get($data, 'username'); // your form should have username field
$password = \FluentForm\Framework\Helpers\ArrayHelper::get($data, 'password'); // your form should have password field

if(!$username || !$password) {
wp_send_json_error([
'errors' => ['Please provide username and password']
], 423);
}

$user = wp_authenticate($username, $password);
if($user && wp_check_password($password, $user->user_pass, $user->ID)) {
wp_clear_auth_cookie();
wp_set_current_user($user->ID);
wp_set_auth_cookie($user->ID);
/* user is not logged in.
* If you use wp_send_json_success the submission will not be recorded
* If you remove the wp_send_json_success then it will record the data in fluentform
* in that case you should redirect the user on form submission settings
*/

$redirectUrl = home_url();


wp_send_json_success([
'result' => [
'redirectUrl' => $redirectUrl,
'message' => 'You are logged in, Please wait while you are redirecting'
]
]);
} else {
// password or user don't match
wp_send_json_error([
'errors' => ['Username / password is not correct']
], 423);
}
}, 10, 3);