The US has a set of values that is really hard to find elsewhere. For example, it is technically wrong to refer to the President as His Excellency, since that, according to their founding fathers, is against the republican principles that no man should be over another. The correct form of addressing the president is by using the vocative Mr. President, and introducing (or referring to) him simply as the President of the United States. No need to say that can cause some very awkward moments in the international arena, where it is customary to refer to any republican Head of State as Excellency. Nowadays, the presidency accepts to be treated as Excellency by the international community, even though there were cases in which the Presidents have corrected Americans when addressing them correctly. There was a case with George Bush correcting a reporter after being called 'sir'. He said the correct term was Mr. President, and that it was a similar situation to calling the Pope anything other than Your Holiness.
There was also the issue with the State of the Union Address. Originally it was simply delivered by the President, since a speech was too "monarchical", going back to the old Speech from the Throne. Obviously, this has changed at some point in the 20th century.
The next issue is the case of an insignia. Monarchs have used crowns to show they are the ones they claim to be, for centuries. Of course, in most republics, the idea of a president wearing a crown was seen as ridiculous. There were other symbols that were seen as insignias as well. Collars and chains of office have been worn by state officials for centuries, as well. In fact, the main insignia used by the Russian President is precisely a chain of office: a collar with the Russian Coat of Arms. Judges in Portugal wear a robe and a collar with a huge medallion. In Peru, the judges don't even wear the robe anymore, just the medallion.
The other important symbol was the sash. Originally it was used by higher Generals, in the Army. Then it became more associated with honorific orders. Most Orders have a sash as the main distinctive of their greatest rank. The President of France, for example, often wears the sash of the Legion of Honor with his white tie, when he has to. In Portugal, there is a distinctive presidential sash, called the 'Band of the Three Orders', which represents the Head of State (it was used by kings, when Portugal was still a monarchy).
In some republics, the sash evolved into some sort of "presidential crown", used to identify the Head of State. The first country to do that was Chile, I think. Then it spread through the entire Hispanic world in the beginning of the 19th century. Brazil only adopted the custom in 1910. We were a monarchy in the 19th century, and the Emperor used a crown. Nowadays, the custom is common all through Latin America, but it has also spread to several other regions. Croatia and Lebanon have their own presidential sashes, for example. Many African nations, as well. Here are a few examples (Croatia, Mexico, Panama and Argentina - images from the Flags of the World website):
And the custom is not used only for Presidents in Republics. In Belize, for example, both the Governor General and the Prime Minister are represented by official sashes, respectively:
The President of Suriname wears a collar that is in-between a Presidential Sash and a Chain of Office:
And finally, the Brazilian Presidential Sash costs a few thousand dollars (the last time I heard about it, it said the new version would cost around 22 thousand dollars). What really represents the President is the golden medal medal. The sash itself changes a lot, but the medal in the bottom of it, has been the same since the first sash:
Anyway, on topic, I always found it weird that the American president has no distinctive sash. It would be nice seeing Obama with a 13-stripe sash, and the US Coat of Arms in the center What do you guys think? (and for the record, I don't really expect many people to actually bother answering this thread, but it is worth asking)
PoFo ethnic party statistics: http://www.politicsforum.org/forum/viewtopic.php?f=4&t=8&p=14042520#p14042520