Nearly 3 years later and it would appear Fallout Shelter is finally making its way to the PlayStation 4. This news comes from a leaked trophy list for the game that has shown up on Exophase, a website that lets you track your PSN, Xbox, Steam, Origin, and Blizzard activity (including trophies and achievements).
Notably, if this leak is real, Fallout Shelter also appears to include a platinum trophy — which I imagine is great news for all you hardcore trophy hunters out there. Outside of that, however, all the trophies are pretty much the same as the Xbox One achievements.
Source: Destructoid Fallout Shelter trophies spotted online