Koi can not recognize the owner. The brain of fish is not as developed as that of mammals. Koi do not have the ability to recognize their owners. Koi will come forward when the breeder appears near the fish tank, just waiting for the food fed by the breeder, which does not mean that they can know the owner.
How long will the koi not recognize the owner. However, if you feed the koi regularly every day, after a period of breeding, they will develop the habit of eating regularly. When the host wants to feed the food, they will swim with the host and wait for the host to feed the food. This is a kind of conditioned reflex formed by them after a long time of breeding, not the performance of recognizing the owner. Koi will follow the breeder when they feel the breeder is near the tank, just because of this conditioned reflex.
Koi's brain is not developed enough for them to have the ability to recognize their owners. So no matter how long they are kept, they can't know their owners. But if they are raised for a long time, they can also develop the habit of eating regularly.