What do people mean by this exactly? I've heard some people make the case that the exact boundaries between ethnic groups can be socially constructed, which makes sense to me (like how one draws a hard line between two groups which have some admixture).
But the idea that ethnicities have nothing to do with biology and is a purely social phenomenon sounds like a lay-person's misunderstanding of the above claim. For example, it should be obvious that using a sperm/egg donor can lead to a child of a different ethnicity even if the socialization is kept the same (which you can't necessarily do perfectly in reality, but still).