Hi,
The cart is supposed to stay. It's the case if you have a HikaShop on a Joomla without anything else. However, based on what other extensions you have, it's possible that the user session from before the registration is cleared before HikaShop can move the cart from the guest session to the logged user session. However, it will be difficult to sort out what is causing the problem.
Another possibility is that you're opening the activation link with another browser than the browser you're using to register. In that case, the cookie identifying the user session on one browser is not available on the other browser and thus the cart is lost when you click on the activation link.
A simpler approach might be to turn off the new user validation system of Joomla via the options page of the Joomla user manager:
docs-next.joomla.org/help/users/users-options/#user-options
Hopefully, it will circumvent the problem, with the added benefit that users won't have to validate their registration, leading to a smoother checkout experience for your users.