All people living in the USA are 'limited' by our supreme "Law of the Land".
You just agreed that to race at a public track, the jockey club could [constitutionally speaking] limit what he 'calls' his horse. - You can't have it both ways.
It's a private track, open to the public. There's a difference.
Fine. - We agree then, that [constitutionally speaking] at a private track, open to the public, racehorses names can be limited.
Thus, - the Constitution puts limits on people, as well as on governments.
It really has nothing to do with the Constitution. Government is not involved, only the private sector and how they chose to allow or disallow whatever. The primary underlying reason is money. They don’t want to “offend” any current or potential “customers” by having a horse named something that would be considered “offensive” or disrespectful. The “market place” puts its own restrictions on things outside of the purview of any government constitution.
Now if the government (of the city, county, state, or federal) had made a law, ordinance etc. that prevented you from exercising that “freedom” to name your horse, cat, dog or car, something (they, government bureaucrats, etc.) deemed “offensive” THEN I would have a “figantice, hugh and series” problem with it!