DjBeau We're always happy to help here
A web app is basically a bookmarked website that looks like an app on your home screen and looks like an app when you open it, but its actually just a website, which means it is isolated from other apps on your phone (normal apps can communicate with each other).
As for privacy with a VPN, they dont block cookies, and regardless, there are many other ways to identify a user. See for yourself here. If you never log in or provide any personal information, they cant associate your searches with you, but they can associate them with each other.
If you prefer google search results, consider alternative frontends like Startpage. They'll get you the same results, but wont send google all your fingerprinted data and cookies.