Create page (not the post type) dynamically

You need to register a ‘rewrite endpoint’ using add_rewrite_endpoint():

<?php
add_action(
    'init',
    function() {
        add_rewrite_endpoint( 'play', EP_PERMALINK );
    }
);

Now, after flushing permalinks, in whatever template or code you’re using for your products you can check this to determine whether the /play version of the URL is being accessed:

<?php
if ( false !== get_query_var( 'play', false ) ) {
    // True for mydomain.tld/product/play-and-win-samsung-galaxy-s22/play
} else {
    // True for mydomain.tld/product/play-and-win-samsung-galaxy-s22
}