"A horse apiece." Where did this phrase come from?

An excellent question. IThis site suggests that it's an equivalent for "Six of one, half a dozen of the other":

https://www.waywordradio.org/etymology-of-a-horse-apiece/

A Horse apiece is used in the midwest when playing a game called Bar Dice. When the game comes down to the last two people, it is the best out of three. When both of the last people have one win apiece it is than considered a horse apiece, and the two people have to play one more round.

Bar Dice rules and how to play:
5 dice in a cup

Each person playing gets three shakes.
--Aces are wild
--Must have an Ace to start
Object to the game is to have the most of a kind (remembering aces are wild, and must have an ace to start holding)
--Person with the highest set is out.
--If two people tie -- everyone re-shakes
Game continues on until it gets down to the last two people where in turn it's the best out of three (again where the phrase a horse apiece)
-- The last person in, aka loser, than must buy everyone who was playing a shot or beer.

(Ex. If someone has 2 aces and a five. That person now has 3 fives. Next person has 2 aces and 2 threes, they now have 4 threes. The person who has the 4 threes is out, and the person with 3 fives continues to play.)