Vivien,
No one really knows.
Piss like a horse......is an older expression, but not very common, from a time before the automobile when everyone knew about horses
Piss like a racehorse......urinated a lot and urgently. This newer American expression came about as betting on horse races became popular in America. People saw the horses nervously peeing before a race. If a horse did this before the race it was an advantage since it could lose up to 10 pounds. In the 70s trainers started giving a drug called "lasix" to their horses. The drug causes the animals to pee urgently and copiously before a race, often in front of the crowd.
piss like a Russian race horse..... Russian horses were bred for the army during the Soviet period. The result was a breed of Arabian horse that was extremely athletic, intelligent, and good-natured. In the 80s every breeder in America wanted a Russian Arabian, not for racing since Americans already have the fastest horses in the world, but for their other excellent qualities. Russian horses became very, very, expensive. Soon, of course, there were too many of them and the market collapsed as the prices fell.
The expression probably originated in rural America, where people know about horses. A Russian Arabian is synonymous with an exotic, expensive, superior horse. Add that to the fact that Russian sounds like 'rushing" and the alliteration makes a very colorful expression as Daniel mentioned.
Pee like a Russian (rushin) race horse.
Some claim that the expression is negative because Russian trainers (or the Russian mafia) cheating by feeding their horses a lot of water, thus making them heavier, or by somehow preventing them from urinating, thus making them nervous and faster. These horses would then pee a lot and urgently after the race. This is unlikely because Russian horses were not used in America as racehorses, nor would the public get the chance to see the horses after the race.