I see that some websites have a fancy push notification permission request popups like this one
whereas official API documentation doesn't provide any clue how to customize permission requests; and the only request I can generate looks like that
How can I create a less ugly popup?
Click the Allow button on the House of Bots site and you'll see exactly how they do it--they don't.
The pop-up you see is just plain html, nothing special. When you click allow, it opens a new window that asks for permission, and then you get the typical browser window just like in your second image.