This limit was(/is) actually a common problem with HTTP/1.1, and was one of the motivations for HTTP/2. A common workaround is "domain sharding"[1] where you would split your site up between several domains, so that you could get more concurrent connections. Which is important for improving page load time, but doesn't really help with having SSE for multiple tabs (unless you do something crazy like using a randomly generated domain for each SSE connection).

[1]: https://developer.mozilla.org/en-US/docs/Glossary/Domain_sha...