Every child gets half of their genes from the mother and half from the father, but what they get is random. Even multiple children from the same parents are not identical.
If Anna and Alice are identical twins and Ben and Bob are identical twins then children from Anna+Ben and children from Alice+Bob will (on average) have 50% of their genes in common, the same as e.g. multiple children of Anna+Ben. Genetically they are as related as siblings.
Just because the parents are genetically identical doesn’t mean the children will be. Siblings have the same parents and can look totally different, after all. For example my sister and I have the same parents, but she’s white and I’m black. Genetics is a weird thing sometimes. This is the same kind of thing.
A pair of identical twins having children with a pair of identical twins would result in children that look like siblings genetically, despite being cousins.
A pair of identical twins having children with a pair of fraternal twins would result in children that look like they fall somewhere between siblings and cousins. Specifically, they’d look like half siblings staying within the same family. Ie parent A has a child with parent B, and then also has one with parent B’s brother or sister.
A pair of fraternal twins having children with a pair of fraternal twins would result in children that look like cousins genetically, as expected.
Fraternal twins are two children born at the same time, made from 2 eggs, 2 sperm, and had 2 placentas. Identical twins are made from 1 egg and 1 sperm with 1 placenta, it just split into 2 embryos very early on. Fraternal twins just look like normal siblings genetically, but identical twins share the same exact DNA.
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