My guess is
American Pharoah (sic) the horse that recently won the Triple Crown.
A monarch in democracy,
Pharaohs were monarchs, but the U.S. is a democracy.
yet revered without hypocrisy,
He's a horse, not a political figure.
my name shall live on evermore.
Triple Crown winners are rare enough to be famous as long as people keep racing horses.
Under the same reign as the sign of war,
Not sure about this one; I think it's a reference to the zodiac, but I don't see a connection. Maybe a reference to War Admiral, a previous Triple Crown winner?
Could either refer to his jockey also riding War Eagle, a previous contender, or his lineage, as he is a descendant of War Admiral and Man O' War, among other notable thoroughbreds.
and thirty-one's twenty-four,
A possible reference to Secretariat's 1973 win by 31 lengths.
the first to seize a third at three.
His jockey, Victor Espinoza, was the first to ride 3 different Triple Crown horses, winning it on his third attempt.
Totaling thirteen and a little more,
Refers to the how many lengths he won by, over the 3 races. 1 + 7 + 5 1/2.
I became the one to do it, finally;
Referring to being the first Triple Crown winner in 37 years.
I'm now a monarch doubly.
Both from his name and the Triple Crown.
Credit to regan for help on lines 4 and 6, Herb Wolfe on line 7, question_asker, whose comment about LeBron James and Kobe Bryant got me thinking about sports, and Big Black Box, who corrected the name's spelling.