Palm’s webOS did a unified inbox the best way. BlackBerry had a few implementations in the 2000s but the one this article refers to wasn’t really a true unified inbox, in part because the applet support for many of the services was weak (you had to go through weird protocols to send IM messages through BIS/BES whatever). The latter implementations with BlackBerry 10 were OK, but this was something Palm really nailed.
Had to scroll through almost 100 replies before seeing any mention of WebOS and it’s awesome unified mailbox. For the few fortunate to use it it was an amazing piece of tech. Unfortunately it wasn’t enough to push WebOS into the mainstream. Palm was bought by HP who didn’t have much imagination and eventually sold WebOS and the IP for peanuts.