Firefox would be very difficult, because Firefox uses a single process for all tabs. You'd need to write a Firefox extension that either emitted a tab-id as a fake HTTP header, or that controlled the proxy for that individual tab.

Chrome would also be very difficult, because, while it uses different processes for each renderer, all network traffic (today) is sent by a single process. An extension may be able to help, as above.

Internet Explorer /sorta/ allows this, insofar as modern versions of IE attempt to use one process for each tab, and that process does its own network requests. However, IE can interleave multiple tabs into a single process, which it does for performance and functional reasons.