Haha, I can see why you're confused. This stuff will get more obvious over time -- we're in a transition stage between "Hack Reactor is a school" and "Hack Reactor is a network of similar schools, run by different teams". Hooray for awkward teenage years! For a current account (likely out of date by the time you read this) keep reading. This stuff also applies to Telegraph Academy and Hack Reactor Remote.
Begin by reading up over here: I was recently accepted into MakerSquare and Hack Reactor. Which one should I choose?
Job search support: this is much bigger than just hiring day. All partner schools have identical job search support curriculum/process (but are administered by different people). The hiring days for all partner schools are run by the same team.
Perception: I would guess that all partner schools will end up having similar reputations, unless they're in the process of crashing and burning. Talk to recent students.
Alumni: I would guess that they'll end up feeling like one family, but right now they feel like separate families.