The president is always referred to CiC out of context, the amount of times I am watching American television coverage and they say something like 'we are now going live to White House, where the Commander in Chief is taking questions from the media'...
Now relative to the UK, when have you ever heard the PM or the Monarch referred to be such a title? The vast majority of the population wouldn't know who the CiC is or know that there even is such a position in the United Kingdom at all.
With regard to the Prince of Wales he served in the military and is Heir Apparent to the throne and as such will be the Commander in Chief of the British Armed Forces. This idea that the Royal Family just parade around in military uniforms without serving is ridiculous - The Prince of Wales, The Duke of Edinburgh, The Duke of Gloucester, The Duke of York, The Duke of Kent, Prince Harry and the Queen herself have all served in the military and on many occasions have been to war.