Folders with Icons on iOS
First of all, let me just put this out there: any tweak that tweaks the appearance like this is going to require that you jailbreak ;)

While not exactly what you're looking, A3Tweaks has achieved a somewhat similar alternative to what you're describing! It's called "Apex 2 (iOS 7)". It is a paid app, but at $2.99 it's not entirely unreasonable. However, it is slightly different that being just a folder with one icon. Rather than wasting your or my time writing out exactly what it does, what it looks like, or how it functions, you can check out an excellent write-up at iDownloadBlog.

Hope this helps!