Update post terms with custom taxonomy

Yes it is. But you have to know the exact taxonomy’s slug, since colors are not like preserved terms to be generated programmatically. Here is an example of how to do it:

// run our function when a post is published
add_action('save_post','update_my_taxonomies');
function update_my_taxonomies($post_id){
    // Check if the post has a particular taxonomy
    if(has_term( 'blue', 'colors', $post_id )){
        // Assign a term to our post
        wp_set_object_terms( $post_id, 'blue', 'blue_bike' );
    }
}

This will set the blue for the blue_bike if the post has the blue term as its color taxonomy.