I don't believe the horse Princess Haya has anything to do with the Royal living person with the same name.
You are correct, the horse isn't owned by the Maktoums, nor was it bred by them. It was sired by a Maktoum owned stallion and born in Texas, USA.
What's interesting is that under Jockey Club naming rules, Thoroughbred horses born and registered in the US aren't allowed to be named for any living person or celebrity unless that person has given consent. They also don't allow names of historical or "notorious" persons.
The US and Great Britain are pretty strict on this, for example I checked for a "Queen Elizabeth" and the only countries with a horse of this name were Uruguay, Turkey and Argentina. The last year the US had a horse named "Queen Elizabeth" was 1933 and the name hasn't been reissued since. Great Britain had a "Queen Elizabeth II" in 1908, and a "Queen Elizabeth" way back in 1899 and 1859.
So if the name wasn't ruled "generic", either Princess Haya gave consent, or it slipped by, which has been known to happen. If she did give her consent for the horse, thankfully she gave it to a pretty successul one!
For fun, I checked for other some other names, the country which allowed the name and the year the horse was born:
"Sheikh Mohammed": Brazil (1989)
"Fazza": Great Britain (2007) This is a 2 year old that raced for the first time last week. He came in 11th out of 13. He's not owned or bred by the Maktoums either but he was sired by a Maktoum-owned stallion.
"Maktoum": Canada (1981)
"Sheik Rashid": Puerto Rico (1991) (Perhaps this was a tribute to the late Sheikh Rashid bin Saeed who died in 1990??)
I could find no other horse in any of the Thoroughbred registries worldwide named "Princess Haya".