Does WordPress Loop have a function like Shopify’s cycle?

Short answer, no.

But that is a pretty cool feature!

To so something similar in WordPress, you’d need to code some kind of iterator in PHP and have it do the logic for you. For example:

$classes = array( 'first', 'second', 'third' );
$iteration = 0;
if ( have_posts() ) : while ( have_posts() ) : the_post();
    $iteration = $iteration >= count($classes) ? 0 : $iteration;
    $class = $classes[$iteration];
    ?>

    <div class="<?php echo $class; ?>">
        <!-- Whatever other stuff you need to do -->
    </div>

    <?php
    $iteration++;
endwhile; endif;