When I spoke with an official from OFA last year, I was told that the organization didn't think people should be confused. The account's bio clearly states it's run by OFA, and it notes that any tweets actually from Obama will be signed "-bo." (Over at least the past four months, there have been zero tweets signed in that way.) But the bio only shows up when you go to twitter.com/barackobama; retweets of OFA's Obama tweets don't indicate that it's not the man himself doing the tweeting.
So what ends up happening is that people (like the confused, mourning girl at right) think they're interacting with Obama, just as people who respond to @WiIIISmith or whatever think they're interacting with the celebrity. And at no point, it seems, has the OFA used the @BarackObama account to clarify to people that they're mistaken in thinking the account is Obama himself (or even his staff) tweeting the president's thoughts.
Bump highlights tweets from the @barackobama account about Michael Sam and about House of Cards, picked up by various media outlets and reported as statements made by the president, not statements from a twitter account run by OFA, the 501(c)4 that is supposed to be primarily about “social welfare” civic issues but also runs @barackobama and barackobama.com.
OFA’s 501(c)4 status was made possible by the decision inCitizen’s United, and the organization operates under the same rules as a wide swath of opposition 501(c)4s that were given extra scrutiny by the feds. 501(c)4s, in fact, are something Democrats use as a kind of scapegoat even as they rely on their own to try to influence and win elections too.