That is a very good question and one I have pondered myself many times. I think the American admiration of Napoleon derives from a number of sources, vizt.:

1. HE WAS FRENCH AND THE FRENCH ASSISTED THE REBELS IN 1775-1783
Americans have not fogotten the assistance given them by France during the proto-Marxist insurrection known by some as the American Revolution.

2. HE FOUGHT BRITAIN
Just as Americans have not forgotten France for assisting them, they have not forgotten that the British were their enemies during the Revolution and, thus the enemy of my enemy is my friend.

3. AMERICANS, BY AND LARGE, ARE NOT KNOWLEDGEABLE ABOUT BRITISH HISTORY
Particularly the history of Britain during the Great War with the French, 1793-1815, and particularly the history of the British armed forces

4. AMERICANS EQUATE THE FRENCH REVOLUTION WITH THEIR OWN REVOLUTION
A people who rise up in rebellion against a king and later fight Britain ruled by the same king Americans rebelled against, must be worthy of admiration

5. THE AMERICAN ARMY USED FRENCH MANUALS
Around 1800, the US army switched from British manuals to French manuals, particularly for the infantry and artillery. The infantry manuals by which American troops were trained in the War of 1812 were abridgements of the 1791 Reglement and the Reglement was adopted holus-bolus (at least the translation by Macdonald) as the official US manual in 1815. After 1815, Major-General Winfield Scott translated all the current French manuals and published them under his own name as SCOTT'S TACTICS right up to about 1861. At one point he became so tired of translating that he suggested that the US Army adopt French as a language of command which drew forth the response from his life-long enemy, General EP Gaines that "the English language is sufficient for all the ideas that General Scott is likely to have." Same for the artillery manuals which were either direct copies of French originals or abridgements.
In contrast, the USN adopted RN regulations almost complete and both the army and navy adopted British military and naval court martial procedure, because both nations used the Common Law and British court martial procedure was based primarily on it.
Regarding manuals, one point should be made and that is that British manuals were not readily available, and French were.

6. THE AMERICAN ARMY ADMIRED FRENCH STRATEGICAL AND OPERATIONAL DOCTRINE
Mainly as codified by Jomini who was much read in the USA between 1815 and 1861. Dennis Hart Mahan, who taught the theory of war (as much as it was taught) at the US military academy for nearly 40 years stressed Jomini's analysis and exposition of Napoleonic tactics. John Armstrong, who served as an ambassador to France prior to the War of 1812 and as secretary of war during that conflict, wrote a book called HINTS TO YOUNG GENERALS FROM AN OLD SOLDIER, almost entirely cribbed from Jomini. One of the most popular translations of Jomini into English was done by Henry Halleck, the senior general of the Union Army in the Civil War (although his text on military law is derived from British practice). Not that Jomini proved all that useful during the Civil War.

7. AMERICANS BECAME CAUGHT UP IN THE GREAT PASSION FOR NAPOLEON AND HIS EMPIRE IN THE LATER 19TH CENTURY
When the French, not unnaturally after the debacle of 1870-1871, turned to the more distant past both for lessons and for nostalgia, producing copious works on Napoleon and his government, Americans got caught up in it. Several American historian, notably TA Dodge, produced panegyrics on the French ruler and his state. Significantly not many Americans were aware -- even though the language was same -- of the great explosion in British Napoleonic literature which occurred at the same time.

And that brings us up to the First World War when France trained, armed and equipped the American Expeditionary Force (remember Pershing saying, "Lafayette, WE are here!" when he arrived in France. This promoted closer relations between the American and French armies and their respective nations.

I think I will leave it there. Put it all together (and there may be more) and what you have is a great American admiration of Bonaparte and his works when, given that Britain and the USA have a common language, a common law basis, and many other common elements, it would be more logical for them to admire the British. But what has logic got to do with it.

dg