non-app wish list using customer.tags
To make this wish list work, you'll need to insert wish_list_product_snippet.liquid somewhere in your product page and wish_list_page_snippet.liquid in a page with the handle wish-list. You'll also need to create a smart collection called "All" (make sure it has the handle "all") that includes all products with price greater than 0.
You can also add a link to the Wish List page on the customers/account template, but it's not required. There's no in-line styling so they should pick up your product and page template styling.
The wish list works by adding the product id to the customers account tags. To remove the item, another tag is added with an "x" preceding the product id. Re-adding the item to the wish list adds another "x". At this point there are three tags for the one product. If a customer went crazy with the wish list, adding and removing and re-adding products, it could fill up their customer tags quickly. If there's any issue with their account or interference with an app (like a wholesale app), I'd look first to their tags.