Unified push allows apps that support it to 1) receive push notifications, 2) without relying in Google, and 3) without having each individual app running in the background & draining the battery.
From what I can tell, Jami shouldn't actually require unified push. It says it can use Play Services if you have that. If not, it sounds like it should also be able to run in the background and keep a socket open. That might (or might not) consume more battery, but will be much simpler than setting up a Unified Push service.