When you create a product discount via the Easy Upsell app, it auto-creates a "Promotion" in your BigCommerce store using the BigCommerce API. (When you create a product discount within the app, the promotion also appears within BigCommerce under Marketing > Promotions. The name of the promotion in BigCommerce will be "Easy Upsell Promotion for "Upsell Name")
Therefore, if customers within your store purchase the same combination of products outside of an upsell, they could still receive the discount. Additionally, product discounts may stack on top of each other if you select the option to "Allow this discount to combine with other promotions on my store."
There is not currently a way with the BigCommerce API for the app to create an exclusive discount that can only be used in the pop-up.
However, if you want to keep the product discounts exclusive to the app, there is a simple workaround:
In BigCommerce, create a duplicate version of the product you wish to discount in the upsell pop-up. Give the product a unique name that you will remember (ie "XYZ Widget Discount Offer").
Disable visibility for this product in BigCommerce. This will prevent it from showing up on your website, but it can still be used in the upsell pop-up and purchased that way.
In the Easy Upsell app, create the upsell offer and add the duplicated product you created as the upsell product (finding it using the unique name you gave the product in BigCommerce).
Following these steps will allow you to offer custom discounts to customers in the upsell pop-up, without customers being able to stumble upon the discount organically on your BigCommerce storefront.