Add new post in existing categories using wp_insert_post

post_category is only for Core Categories. Use tax_input instead:

$new_post = array( 
    'post_title' => esc_attr(strip_tags($title)),
    'post_content' => esc_attr(strip_tags($description)),
    'post_type' => 'proiecte',
    'post_status' => 'publish',
    'tax_input' => array(
        'categorie' => array(17,16)
    )
);
$post_id = wp_insert_post($new_post);

It may be clearer to other developers, and yourself in the future, if you use something other than categorie – perhaps projecte_category – just so it’s clearer you’re using a custom taxonomy and not misspelling Category.