As a Goldman Sachs financier, Mr. Scaramucci may have the least communications experience of any of the president’s aides. But his never-give-an-inch defense of Mr. Trump since before the election has endeared him to the man in the Oval Office. He was a frequent presence at Trump Tower during the transition, often appearing before the throng of reporters in the lobby. And since the inauguration, he has been all over television, defending Mr. Trump and his agenda.