You cannot run two instances of app with same package name because when you try to install it out of the box because the package installer checks if an instance of the package you are trying to install is already present and if present asks if you want to update the existing app.
You can however change the source code of an existing app to fool the system that it is some other app with a different package name by cloning the source code and have a different package name. Reference link herehere
Also there's another way by making use of the special Facebook lite app.
Facebook has provided a Lite app which is supposed to work for low-data-speed devices consuming less memory.
You can install one Facebook native app and another Facebook lite app and use multiple accounts. So you'll be able to get notifications from both the apps.
NOTE: This is possible because Facebook-app and Facebook-lite are two different apps (with two different package names com.facebook.katana
and com.facebook.lite
respectively)