Snapchat takes two approaches (that I know of): First, they don't stop it, but the messager is supposed to be notified when it happens. They know this doesn't work and can't work, though; avoiding screenshot detection isn't even considered a reportable bug. (see https://hackerone.com/snapchat )

Second, they have a legal barrier, in that using a third-party client to communicate over Snapchat violates their terms of service.

Notably, neither of those makes the messages any more ephemeral. This makes sense, because stopping people from recording messages that you send them is not an achievable goal. Barring some sort of supernatural influence, it's no more achievable for Instagram than it is for Snapchat.