Drupal - Get username in custom user twig template

The display name is not a field you can configure in {{ content }}. You can get it directly from the user entity:

{{ user.displayname }}

Reference for the php method: AccountInterface::getDisplayName

{{ user.name.0.value }}

should give you what you want.

The preferred way is inject the username as a variable in a preprocess function. To do so, tweak that in your theme's .theme file

<?php

/**
 * Implements hook_preprocess_user().
 */
function yourtheme_preprocess_user(&$variables) {
  /** @var User $account */
  $account = $variables['elements']['#user'];

  $variables['username'] = $account->getDisplayName();
}

then in your user template you can use that as {{ username }}.

Edit:

To get the full URL:

• In your template {{ path('entity.user.canonical', {'user': user.id}, {}) }}
• In yourtheme_preprocess_user(): $variables['user_url'] = Url::fromRoute('entity.user.canonical', ['user' => $account->id()])->setAbsolute()->toString();