As far as I can tell, there is no software that integrates multiple chat protocols and email. However, if you’re okay with having a separate email application, you may want to have a look at a multi-protocol messaging client like Pidgin or whatever it is that KDE uses these days (used to be Kopete and Telepathy).

EDIT: Scratch that. Kopete and Telepathy don’t have Matrix support. Pidgin still has it, though.