Not every website responds to this in the same way. Collection of techniques known as Progressive Web App (PWA) cause the mobile browser to behave differently than just loading it in the browser normally. The website needs to be configured to take advantage of this. Launching a PWA from the app icon like you demoed instructs the mobile browser to hide its default ui - address bar and control buttons etc - allowing the web app to take over the whole screen giving an immersive mobile app experience. Additional features that allow for custom network and storage interaction can give certain PWA’s offline features functionality as well.
ICYMI 💜
You can make any website accessible like an app..
Only from Safari afaik.
Check out this demo with timechain calendar.com
nostr:npub1pdzn5fqc89a02yx6uynvn2s2hvgc4f2ecshd97h4cjucl5s8u0zqmxe70u 👇 https://nostr.build/av/880df7c140a2552660290a458ba685c2847e9a7ca39f2107b0ff3c52db8d22a3.mov 
Discussion
Oh interesting.. so it seems you did optimize the site then for being a PWA yes?