Yeah, it sure is ironic. In fact it's WAY more than ironic. In actual fact the Commander in Chief has enormous power over the military to appoont and fire flag officers and to direct military operations.

So what does it mean to say it's a "civilian office"? Not all that much.

The president salutes and directs the military. He commands the military. You and Wikipedia want to call that civilian - go ahead and call it civilian but remember: Truman dropped the atomic bomb on Japan and fired Douglas MacArthur.

So he doesn't wear a uniform? Not under UCMJ? What's more important, command and control or clothes?

President Abraham Lincoln was deeply involved in strategy development and day-to-day military operations during the American Civil War, 1861–1865; historians have given Lincoln high praise for his strategic sense and his ability to select and encourage commanders such as Ulysses S. Grant.[20] On the other extreme, Woodrow Wilson paid very little attention to operational military details of World War I and had very little contact with the War Department or with General John J. Pershing, who had a high degree of autonomy as commander of the armies in France.[21] As president during World War II, Franklin D. Roosevelt worked closely with his generals and admirals, and assigned Admiral William D. Leahy as "Chief of Staff to the Commander in Chief".[22] Harry S. Truman believed in a high amount of civilian leadership of the military, making many tactical and policy decisions based on the recommendations of his advisors—including the decision to use atomic weapons on Japan, to commit American forces in the Korean War, and to terminate Douglas MacArthur from his command.[23] Lyndon B. Johnson kept a very tight personal control of operations during the Vietnam War, which some historians have sharply criticized.[24]