Is there a way to fetch the category of a page when using wp_list_pages?

You can get an array of pages using get_pages() and then for each page retrieve categories using wp_get_object_terms. Your code may look like this:

$args = array(
   'post_type' => 'board-meeting',
   'post_status' => 'publish'
 ); 

$pages = get_pages($args); 
if ( ! empty( $pages ) ) {
    echo '<ul>';
    foreach($pages as $page){
          $cats = wp_get_object_terms( $page->ID,  'category' ); // Array of categories
          $cat="";
          if ( ! empty( $cats ) ) {
              if ( ! is_wp_error( $cats ) ) {
                  $cat = $cats[0]->slug; // slug of 1st category in the array $cats
              }
               // slug of category used as CSS class name for anchor tag
               echo '<li><a class=". $cat." href="' . get_permalink($page-ID) . '">' . esc_html( $page->post_title ) . '</a></li>'; 
           }
    }
    echo '</ul>';
}

You may adjust the code to your needs.

I hope this helps.