I’m trying to display product categories on each product in the list

Product categories in Woocommerce is a term, so you can get the categories with get_the_terms function:

echo '<h3><a href="' . get_the_permalink() . '">' . get_the_title() . '</a></h3>';
echo '<span style="display:none;" class="prodcat';
$categories = get_the_terms( get_the_ID(), 'product_cat' );
foreach ($categories as $category) {
    echo ' ' . $category->name . '';
};
echo '"> </span>';