Nick here from Shopify. Good question!
Thanks for sharing the screenshot. It is very helpful to figure out what you are looking for exactly. We are unable to say whether or not the store you linked is even using Shopify (let alone the exact app) for privacy and security reasons, but I have an idea from the screenshot of what you are looking for. It looks like it is an upsell app of some kind. I took a look through the Shopify App Store and from my experience, I found some of the most popular upsell apps which allow you to create a pop up before checkout like you are looking for which you can see below:
All of these apps have slightly different features but accomplish the same general principle. You could also look into having something custom made for your store. While it would be a much higher upfront cost, it would be tailored to you specifically. It could be worth reaching out and talking to some Shopify Experts about this from the Expert directory here, but hopefully one of the apps resonants with you and you can go down that route.
Hope this helps!
All the best, Nick