Each eye of the potato tuber produces its own plant. Yes you have a plant for each eye. Each one is genetically the same since it is vegetative reproduction. I have experimented with this and find that if I cut the tubers to two eyes per piece, I get larger potatoes than when I plant a whole potato and have numerous plants grow. Oh, I get loads of small tubers when I plant a whole potato.
I don't know that you can remove any of them with total success as the eye will still want to send up a shoot. It is best to cut your seed potato to one or two eyes per piece. I think I would just let it grow. Give it plenty of water.
This one had two eyes on the seed tuber. See how much bigger the tubers are?