How to update 8.0.x to 8.1.x with composer?
I see, you directly get drupal/drupal. I don't think that's actually supported, you still get a vendor folder from git and the composer.json isn't properly set up. I'm using composer command from this site:

composer create-project drupal-composer/drupal-project:8.x-dev myproject --stability dev --no-interaction


Regarding update, the trick seems to be the following argument to composer require:

composer require --update-with-dependencies "drupal/core:8.1.*"


That updated drupal/core and all dependencies it has.