Hello,
Sorry but there is no such option to show the product SKU on the shop page or product category. It’s possible by additional PHP customization. You can use WooCommerce hooks if you have PHP skills https://www.businessbloomer.com/woocommerce-visual-hook-guide-archiveshopcat-page/ to implement this or submit customization request to our customization team here because our default support does not cover additional PHP customization, sorry.
Regards