The Iron Heights penitentiary was a recurring location in the Flash as it holds many of Central City's worst (and for while wrongly held Henry Allen) however dialogue in in Arrow's fourth season says criminals from Starling (or Star city, whatever) are sent somewhere also called Iron Heights- aren't these cities at least a hundred miles apart?
Both places definitely have a prison called Iron Heights, but I doubt it could possibly be the same place.
In Arrow, S01E04, Jason Brodeur is complaining to his lawyer about Laurel trying to send him to prison:
BRODEUR: She's going after me!
ATTOURNEY: She'll wanna meet with her client after what happened in court today. We have friends up in Iron Heights. Prison can be a dangerous place.
This shows that the prison is close enough that Laurel can get up there to visit clients as part of her normal work schedule. There are, in fact, multiple cases where Oliver's team goes up to Iron Heights on short notice. So, that prison must be relatively close to Starling City.
Similarly, In The Flash S01E02, Barry is arguing with Joe and says:
BARRY: I'm not your kid, Joe. And you're not my father. My father's sitting in Iron Heights.
We also know that Barry once ran away to visit his father in prison, long before he got his speed; we also know that Joe visits Barry's dad a few times on his own. So that prison must also be within driving distance of Central City.
Finally, as we've already discussed in this related question, Oliver has explicitly placed the two cities about 600 miles apart; in addition, in the Arrow episode "Legends of Yesterday", the team requires use of Felicity's private jet in order to get from Star City to Central City fast enough to stop Vandal Savage. So clearly, those two cities are not within a few hours drive of each other.
Thus, it seems incredibly unlikely that they are talking about the same prison; I suppose it's feasible that the prison is literally halfway between the two cities, and they routinely drive 10 hours round trip to go there, but that sounds pretty far fetched.
It's the same "Iron Heights" in both shows. The reason is simple sloppiness on the part of the writers. In the early days of production of Flash, they hadn't entirely worked out some details like how far apart Starling and Central cities were, and they'd already established both cities used the same jail as a nod to continuity. This made it on the air before anyone connected wit the shows realied that didn't make any sense. Thus it's an accident they can't really fix, but which we're just kind of not supposed to notice.