"Jesteś moją trzecią córką."
Translation:You are my third daughter.
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Ordinal numerals work basically like adjectives, so grammatically "This is my third daughter" isn't different from let's say "This is my oldest daughter".
That means that they do not affect the grammatical case in any way, there's no Genitive involved. In our sentence here "moją trzecią córką" is in Instrumental due to "Jesteś", in yours it should be Nominative ("To moja trzecia córka").