Even though Elizabeth is refered to as a cousin of Victor, it is clear from her very first introduction into the family that she is actually so much more. You might find it useful to analyse the final paragraph of the first chapter of Victor's account of his story in order to find the answer to your question. Note what he says about when Elizabeth was "given" to him as a Christmas present by his mother:
On the evening previous to her being brought to my home, my mother had said playfully--"I have a pretty present for my Victor--tomorrow he shall have it." And when, on the morrow, she presented Elizabeth to me as her promised gift, I, with childish seriousness, interpreted her words literally, and looked upon Elizabeth as mine--mine to protect, love and cherish. All praises bestowed upon her, I received as made to a possession of my own.
Victor goes on to say that she was "more than sister" and that she was exclusively his possession until his death. Therefore we can see that from the moment Elizabeth entered the Frankenstein family, it was clear that Victor felt only the most strong and powerful bond between him and her.