If you've created a menu item that should only be visible to tagged customers, but it's still visible to others, it will be for one of a few reason.

1: The tag hasn't been added, to the customers account, in the Shopify admin portal. As you can see below, you need to add a tag to the customers account, we've used 'gold' in this example:

 

2: The tag you're using on the customers account, isn't the same as the tag you're using on the page you're trying to hide from non tagged customers. If you example your tag is:

gold

And you've added:

Gold

To the menu items tagging section, it won't work as tags are case specific. You need to ensure both are identical

3: when entering the tag to the menu tagging section, you didn't hit enter. if you don't hit enter before saving, the tag won't be added and won't be saved. It needs to look like the below before saving: