Set default option in dropdown of WP_Query?

(Revised answer)

If I understand it properly, try these:

1. Define $the_post for each “property”:

   <?php $the_post = get_post(); ?>
   <div class="property-details">
       <h2><?php the_title(); ?></h2>
       ...
   </div>
   

2. Then change the drop-down code to:

   <select name="propname">
       <?php
           $found = wp_list_filter( $the_query->posts, [ 'ID' => $the_post->ID ] );
           if ( empty( $found ) ) :
           ?>
               <option selected><?php echo get_the_title( $the_post ); ?></option>
           <?php
           endif;
   
           if($the_query->have_posts()){
               while($the_query->have_posts()) : $the_query->the_post(); ?>
                   <option<?php selected( get_the_ID(), $the_post->ID ); ?>><?php the_title(); ?></option>
               <?php endwhile;
           } wp_reset_query();
       ?>
   </select>