Can your penguin have children on Club Penguin?

No. In some virtual games you can, but not Club Penguin.

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If you would like, you can probably be able to "pretend" to have families living in your igloos, with some users being couples and others being your spouse or parents.

But this would be much of a nonsense..

The closest thing to having a family in Club Penguin would probably be your pet puffles, since you have the responsibilities of feeding them, bathing them, loving them and playing with them.