Hi,
It's possible that your template has a view override for Joomla's registration form which prevents the plugin from adding the captcha on the page.
So you could check whether you have things in the templates/YOUR_TEMPLATE/html/com_user/ folder via FTP. If so, rename the folder and see if that helps, if so, rename the files one by one in the folder to see which one is causing the problem.
A simple way to circumvent the problem, since it works fine with HikaShop's registration page, is to enable the Redirect Joomla registration to HikaShop plugin via the Joomla plugins manager. That way, you'll only use HikaShop's registration form. This has another benefit as the users will be able to provide their address directly at the registration step making the checkout more seamless for them later on when they are ready to purchase.