Is it possible to show multiple (custom) post types at the wp homepage only by using pluging code?

Something like this should achieve your goal. You may need to remove the is_main_query() check depending on where you want this to show up though.

add_filter('pre_get_posts', 'projects_are_posts');
function projects_are_posts($query) {
    if (is_admin() || !is_main_query() ) {
        return $query;
    }
    $types = $query->get('post_type');
    if (!is_array($types)) {
        $types = array($types);
    }
    if (in_array('post', $types) && !in_array('projects', $types)) {
        array_push($types, 'projects');
        $query->set('post_type', $types);

    }
}