Hello, @David,
1. For the Single product page, if you are using the Single Product Built Layout, then you have to go to XStore >> Theme Settings >> WooCommerce >> Single Product Builder >> Product Meta and turn OFF the Tags: https://ibb.co/sFJxWDd
2. For the Shop page, those are not the tags but the categories, you have to go to XStore >> Theme Settings >> WooCommerce >> Shop >> Product Style >> Product Content Element and disable the elements that you do not want to show on the Shop page: https://ibb.co/Df4KNx0
Regards 8Themes Team.