Get access to all terms associated to each post that the wp_query loop displays

If I understood you correctly, you are looking for get_the_terms(), which..

Retrieves the terms of the taxonomy that are attached to the post.

You could for example have helper functions like these,

// functions.php
function my_sports_post_terms( int $post_id ) : array {
  $taxonomies = ['basketball', 'volleyball', 'baseball'];
  $terms = [];

  foreach ($taxonomies as $taxonomy) {
    $terms[$taxonomy] = my_sports_post_tax_terms($post_id, $taxonomy);

  return $terms;

// Force array return value as get_the_terms() may not always return an array
function my_sports_post_tax_terms( int $post_id, string $taxonomy ) : array {
  $terms = get_the_terms( $post_id, $taxonomy );
  return $terms && is_array($terms) ? $terms : [];

And then use the first one in the Loop like so.

// some template file
while ($query->have_posts()) {

  $terms = my_sports_post_terms( get_the_ID() );

  // code..