How to exclude post-tag taxonomy for a custom post type?

Try this to unregister the post_tag taxonomy from the Custom Post Type.

function wpdocs_unregister_tags_for_posts() {
    unregister_taxonomy_for_object_type( 'post_tag', 'post' );
}
add_action( 'init', 'wpdocs_unregister_tags_for_posts' );

Update the 'post' to your custom post type.

Docs: https://developer.wordpress.org/reference/functions/unregister_taxonomy_for_object_type/

Also, you’ve defined the following twice in your $args

'taxonomies'          => array('product'),
'taxonomies'          => array(),