Twins are not acceptable in the community because having identical people would be confusing.
In The Giver, when identical twins are born, they are weighed and the smaller one is released. Release is a euphemism for euthanasia, which means that one of the twins is killed by lethal injection. Release is very common in the community. There are three reasons for release: punishment, the elderly, and newchildren.
When Jonas comments to The Giver that the newborn twin is going to be released, he does not expect Gandalf’s reaction.
"I wish they wouldn't do that," he said quietly, almost to himself.
"Well, they can't have two identical people around! Think how confusing it would be!" Jonas chuckled. (p. 19)
Jonas explains why, but his explanation does not satisfy The Giver. He needs to show Jonas the release. He wants to see one, and it is not at all what he expects. He thinks that his father will make the twin clean and comfy and then the baby will go to live and grow up in elsewhere. He has no idea that his father is a murderer.
The murder of the innocent baby is the climax of the book, and it is an emotional event for Jonas. His world comes crashing down on him. He falls apart. He cannot go home. He cries, and he is angry and frustrated. He and The Giver make a plan then and there that Jonas should leave, and release the memories to the community. The Giver explains that his father killed the baby because he doesn’t know better. The people know nothing.
"I said it because it's true. It's the way they live. It's the life that was created for them. It's the same life that you would have, if you had not been chosen as my successor." (Ch. 20)
The only way to change this is to release the memories back to them. Jonas is the only one who can do that, by leaving. This careful plan goes awry though, when Gabe is scheduled to be released, and Jonas has to escape early.