Whether deliberate or not, the chronic assignment to the President of this title is a method for training the citizenry to conceive of our political leaders, especially the President, as someone whose authority is naturally and desirably expansive and absolute. He's supreme. It converts civilians into soldiers and Presidents into supreme rulers. It's no surprise that this is the shape our government has now taken; this phraseology both reflects and helps to enable the transformation of the President into an unaccountable, virtually omnipotent figure.
Worse still, to equate "the President" with "our commander in chief" is to depict the U.S. as a state of endless war and pervasive militarism. Even in the limited sense that the Constitution uses the term ("Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States"), the President doesn't always wield that power, but only when those branches are "called into the actual Service of the United States."