Terminal Open Safari with Website

The difference is likely to do with the architecture and interprocess communication choices of each browser.

Modern web browsers render pages in separate processes. Apple has a framework called IOSurface that provides a streamlined way for one process to pass an image to another process. This framework was introduced in Mac OS X 10.6, aka Snow Leopard, for the latest iteration of QuickTime.

QuickTime uses IOSurface to offload movie decoding to separate processes. With no decoding to do, the QuickTime Player application is left dealing solely with the user interface and showing images provided by the decoding processes.

I suspect Safari has learned from QuickTime and is using the same techniques. Web pages are offloaded to other processes, rendered, and passed back.

Could Chrome and Firefox do the same thing, absolutely. The challenge is ensuring the thread dealing with the user responds quickly and is not delayed waiting for the rendering to update.

Chrome does use separate processes and appears to use IOSurface on the Mac; this bug talks about improving Chrome's use of IOSurface.

IOSurface is a public framework available to any Mac OS X 10.6+ application. However there is little documentation and it is Mac specific.

This is all conjecture.

Unfortunately, I don't think this is possible.

Safari's push notifications are specific to Safari, and so cannot be configured to open in Chrome. It's a part of individual apps and their implementation of the push notification system.

You haven't specified which Apple OS you are using, but I can talking from my experience of iOS. I don't know if push notifications are a thing in OSX.