Get list of terms that have posts in another term

Try replacing 'operator' => 'AND' with 'relation'=>'AND'

Updated Code Snippet:

$current_color = get_queried_object_id();

$query = new WP_Query( array(
    'post_type'      => 'product',
    'post_status'    => 'publish',
    'posts_per_page' => -1,
    'orderby'        => 'title',
    'order'          => 'ASC',
    'tax_query'      => array(
                              'relation' => 'AND'
                               array(
                                        'taxonomy' => 'pa_color',
                                        'field'    => 'term_id',
                                        'terms'    => $current_color,
                                    ),
                                    array(
                                        'taxonomy' => 'product_cat',
                                        'field'    => 'term_id',
                                        'terms'    => $cats
                                    )
                                )
                        ) );

Reference : https://developer.wordpress.org/reference/classes/wp_query/#taxonomy-parameters ( Multiple Taxonomy Handling )

Edit 1: You can edit above query to return only ids ( ‘fields’ => ‘ids’ ) and use get_terms for those IDs to get the list of categories

get_terms( array( 
   'taxonomy' => 'product_cat',
   'object_ids' => $posts_matching_criteria
);

Ref 1 : https://developer.wordpress.org/reference/functions/get_terms/

Ref 2 : https://developer.wordpress.org/reference/classes/wp_term_query/__construct/#user-contributed-notes

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