While a passport is not absolute proof it can be useful, as are verterinary records. Otherwise a receipt for the horse on purchase and records of photographs and history while you have had the horse are also helpful.
This particular point is where a lot of complaints of theft fall down
One of our horses has a British Hannoverian passport which has an additional and separate "proof of ownership" certificate which has to be signed by the seller and purchaser to change to the new ownership. It would be helpful if all PIOs did the same thing.