Technically it would fit nicely. Unfortunately, it requires the user to explicitly give permissions to show notifications to the website, even if you don't show any notifications. This is a blocker for its usage in many cases.
I think you probably mean web Notification. Although it is always use with Web Push. And also it is generally rate-limited due to its default usage (display a message to user). On the other side, Websocket and SSE talks to program, there is just no rate limit thing exist.
