That is a feature of the browser, and is on the browser to make it not annoying. While it's fair to say you might find it annoying and might not want to see it, others do find it valuable. If a browser feature is annoying you, the correct approach would be to disable that feature in the browser. If the browser doesn't allow you to do this, than it's the browser's implementation that is at fault.
This all assumes the site is using the features the browser makers are telling the sites to use.
The correct answer is that features like push notifications should be opt-in in the browser. Too many of these features exploit the general ignorance of users.
> If a browser feature is annoying you, the correct approach would be to disable that feature in the browser.
Yet when some of us shut off Javascript - a feature that is often annoying, enables many kinds of tracking, and occasionally creates serious security problems - half of HN freaks out. Given that Javascript is necessary for many types of surveillance-as-a-business-model, hostility to the idea of disabling javascript is expected.
Push notifications are still a newer feature, but once that feature becomes entrenched enough that some people's profit depends on the feature, disabling push notifications in the client will be regarded with similar hostility.
This all assumes the site is using the features the browser makers are telling the sites to use.