[tzs]

It only says the mother is Ayn Rand if the comma after mother is taken as making an appositive phrase. There is no later comma to end the appositive phrase, so we'd have to take "Ayn Rand and my daughter" and so it would be saying that his mother is both Ayn Rand and his daughter.

That's not possible so we can rule out the comma making an appositive phrase, and see that it must be a list comma, and we've got no ambiguity.

Putting in the Oxford comma makes it so we could interpret "My mother, Ayn Rand," as an appositive phrase, and we have ambiguity.