nico_login_ng.php

<?php
$client = new \GuzzleHttp\Client([
    'cookies' => true
    ,'http_errors' => false
]);
$client->post('https://secure.nicovideo.jp/secure/login',[
    'form_params' => [
        'mail'        => $mail,
        'password'    => $password
    ]
]);
$response = $client->get('https://seiga.nicovideo.jp/api/user/data',[
    'query' => ['id' => 10908584]
]);

echo $response->getBody();  //NULL

nico_login_ok.php

<?php
$jar = new \GuzzleHttp\Cookie\CookieJar();
$client = new \GuzzleHttp\Client([
    'cookies' => true
    ,'http_errors' => false
]);
$client->post('https://secure.nicovideo.jp/secure/login',[
    'cookies' => $jar,
    'form_params' => [
        'mail'        => $mail,
        'password'    => $password
    ]
]);
$response = $client->get('https://seiga.nicovideo.jp/api/user/data',[
    'cookies' => $jar,
    'query' => ['id' => 10908584]
]);

echo $response->getBody();