Chouby on "[Plugin: Polylang] Menu translation"

In "Primary Menu" do I have to write the name I gave to my menu?

No.

I ended to download your theme. The menus are correctly registered with the name 'categories_navigation_menu' (instead of 'primary' in my code above) in the wile wg-includes/functions/wg-custom-functions.php

So registering the 'primary' menu just added one location per language which is just confusing. The only modification you have to do is to replace:

<?php wp_nav_menu( array('depth' => 3 ) ); ?>

by

<?php wp_nav_menu( array('theme_location' => 'categories_navigation_menu', 'depth' => 3 ) ); ?>

in header.php
And it will work as expected. You then just have to set one menu per language in Appearance->menus.