Gift Registry replaces the shop

You can see this link for a brief documentation of the setting.

But basically, enabling the setting should not remove woocommerce buttons on the shop. The only buttons that should disappear are the “add to wishlist” buttons for users who have not yet enabled the wishlist module.

Hi,
I have the same problem.

I have created a staging environment to better illustrate the problem:https://staging.b4.socialcities.it/test/negozio/

Can you help me?`

Sorry, I got the wrong link.

Is this the correct one: https://staging.b4.socialcities.it/moduloimola/negozio/

This might be a problem with your theme. Try activating a new theme and see if the problem goes away. Also deactivate all other plugins apart from the wishlist plugin and woocommerce to rule out a plugin conflict.