Can we upgrade a site directly to latest version (without an extra fresh site)? As far as I understand, the only way to upgrade Drupal (core-only) sites today is with having two sites: One is the Drupal 7 site with its own folder, and another which is a fresh Drupal 8 site in another folder; We then enable the core-based Drupal 8 migration modules in the fresh Drupal 8 site and with them Migrate-upgrade the Drupal 7 site into that fresh site.

Is there a way, maybe with Drush or with Drupal console, to actually Upgrade the existing Drupal 7 (core, or core&amp;Ctools&amp;Views) site directly as it is, without actually opening a fresh Drupal 8 site, besides it?

In other words, can we use the Console or Drush to update an existing Drupal 7 site directly, without any kind of migration?
No. 

The old method of upgrading Drupal through update.php has been dropped in Drupal 8 in favor of the Migration framework &amp; UI. The idea here is to provide a clean way into a new database schema without any problems of update.php trying to accommodate many diverse scenarios (as it was prone to do with moving complex sites from D5 to D6 &amp; D6 to D7).

D8 has some significant changes to its architecture; trying to patch code &amp; alter the DB schema for a D7 site to move it to D8 could be fraught with problems. It could theoretically be done, but you'd probably spend more time doing that than to just devise a content / custom code migration path.

For more information read Upgrading from Drupal 6 or 7 to Drupal 8 on drupal.org.