Trump didn't "grant himself" authority to step up deportations, it's inherent in his role as head of the Executive Branch, enforcing the immigration laws. [Trump just getting started with immigration raids | The president granted himself sweeping authority to step up deportations, and he's poised to use it, By Seung Min Kim and Ted Hesson, February 13, 2017] Illegal immigration is illegal, and attempts to make it legal—Amnesty legislation—have been rejected decisively by a Congress that was informed by the electorate that if they passed Amnesty, they wouldn't be reelected.
He pointed out that Eisenhower didn't have to pass any laws.
Being an illegal alien was already illegal. All it took was for the President of the United States to decide that something should be done, and then do it.
By contrast, Obama granted himself sweeping authority to stop deportations, something which was obviously illegal, and which we argued was impeachable. (See James Kirkpatrick's If The GOP Leadership Believes Obama’s Amnesty Is Unconstitutional, Why Can’t They Even CONSIDER Impeaching Him?.)
And Obama didn't run on a promise to do that—he told illegals he wasn't king, and couldn't do that. Then he changed his mind.
Trump, by contrast, is keeping his election promise.