You appear to be writing a custom form from scratch using WP_Query to list your options.
To have a sortable select in a form, JavaScript is necessary.
The Pods shortcode has a form="1" option which will output a full form with the admin-defined user interface. It uses React.
WordPress Core enqueues have jquery-ui-sortable, which is a way of making elements sortable. It would not populate a select element; the form would need to be serialized with JavaScript.
Chosen by Harvest is a popular jQuery plugin for turning a select into a searchable UI. It is not sortable.
SortableJS is an example of another sorting library. Like jQuery UI, it creates proxy elements separate from a <select>.
Custom sortable forms require custom logic to submit IDs of relevant selected elements in the correct order.
See WP_Query if you’re looking to search or sort on the PHP side, or PHP usort() if there’s some sorting criteria desired on the server side that’s not provided by WP_Query.
See pods()->save() if you’re looking for a quick way to save the relationship IDs separate from update_post_meta() or add_post_meta()