List Custom Taxonomy Widget

I'll save you the hassle of downloading the plugin and digging through its source code. Just use the wp_list_categories() function in your theme (or plugin) to display custom taxonomies as a list, and use wp_dropdown_categories for a dropdown (this one does require some supplementary html). This plugin is essentially a widget UI shell for these functions. If you want to use the plugin's code for it, go right ahead, but it really does just widgetize those functions.

There is no (need for a) settings page. Simply go to Appearance -> Customize -> Widgets, select the widget area where you want to add the widget, and add the List Custom Taxonomy widget.

If you'd like to do something outside of the configuration options of this plugin, look at the WordPress Codex documentation on the function this plugin implements, wp_list categories. If this function can do it, the plugin can do it with minor edits, just look in list-custom-taxonomy-widget.php in the plugin editor.