Curiously, this namespace issue is solved in other sporthorses with microchips and horse passport numbers. I suppose it's the betting that causes them to have to make rules.

The convention in European sport horses is typically the registered name uses the first initial of the sire (so a horse named Donnerhall's decendents all have D names, and if you have the name and breed, it's like designer advertising for what line they're from) with the exception of the Lusitanos and Spanish PRE's, which use names where the first letter is assigned to the year of their birth.

So a Portuguese or Spanish horse named "A"ndro would have been born in 2004, and a horse named "E"duardo would have been in 2009, and it's a useful convention you can know a horses age from its name. Their studbooks are relatively narrow with fewer foundation stallions and breeders.

Thoroughbreds also have microchips in their necks now, but those solve a different problem. Microchips, and lip tattoos in previous years, ensure that the horse being examined is a particular horse registered at the Jockey Club. However, no one wants to root for or bet on a horse identified by a long random number. So, printable and pronounceable names are still used.