Yes, although I have heard of some unscruplous people applying for duplicate passports to change horses names and ages, this obviously can't happen if the horse is freezemarked or microchipped as the microchips are registered on the passports. I looked at a horse recently for someone to buy, checked its passport, vet checked passport at time of vetting and it had another passport with another issuing agency, so the horse for some reason had 2 passports, in this scenario you have to send the passport back to the agency that has the microchip registered and they will normally sort the problem out for a small fee.
Horse my friend bought had 2, we just sent them both off to one of the issuers and they sorted it out for us.
We think he ended up with 2 because his 1st, main one's issuer address was out of date, so when old owners tried to update it they couldn't. I just google them though and foudnt them. The 2nd one was justa Pet ID one, so we wanted rid of that one because the first was his breed society one.