It's hard to believe now, but Obama's @POTUS handle (and @FLOTUS and @VP) was only created recently in May, 2015. Prior to that, Obama had his own @BarackObama account run by staff, with his own personal tweets signed "-bo." The @POTUS account will pass over to Donald Trump after his inauguration on January 20th, while Obama's presidential accounts will live on as @POTUS44, @FLOTUS44, etc. Trump, of course, is also prolific on social media, particularly Twitter.
You can now search the administration's social media output, though the results are a little scattershot at the moment. The White House points out that there are other ways of tracking Obama's social media impact, though.
For instance, Digital art organization Rhizome is publishing a series of essays explaining internet culture associated with the Obama administration, including the "Thanks Obama" meme, Michelle Obama's Turnip Vine and more. MIT Media Lab developed an interactive tool showing how White House tweet topics changed during the administration. In addition, GIPHY will launch a page that lets the public view all GIFs the White House Shared (above), the Feel Train Twitter bot @Relive44 will republish White House tweets over the next eight years, and Internet Archive is making White House data available on its archive.
Finally, as of yesterday, The White House will let you download its Twitter, Facebook and Vine archives yourself, along with tweets published by President Obama and the First Lady. All told, the administration left social media in pretty good shape for the next president -- lets hope Trump keeps it that way.