Race horse have a special marking mostly on there shoulder to tell which horse is which so all you need to do is find that marking
United States
In the United States, thoroughbred racehorses have a number tattooed inside their upper lip. The tattoo should be one letter followed by five numbers. Thoroughbreds older than 25 years old only have four numbers. You can research the tattoo number on the Jockey Club Registry website to get information about the horse's breeding and race record.
Some thoroughbreds also have microchip implants with identifying information, but you need to contact a veterinarian with a special chip reader to get the information. Microchips aren't required for registration in the US.
If you meant to ask how you can find a retired racehorse to buy, adopt or foster, there are several charitable organizations that place old racehorses in new homes for a small fee. You can get more information through The Jockey Club (the official breed registry), the Thoroughbred Retirement Foundation and special charitable organizations that maintain websites dedicated to finding new homes for old racehorses.
CANTER is a non-profit that rescues ex race horses and adopts them out.
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If its a thoroughbred then the Jockey Club should have his papers. If you don't know his registered name, then check under his upper lip, and there should be a tattoo telling you his number.