פירוש על קידושין 7:1

"A similar going out it comes to exclude". Rabbenu Hananel explained: "The going out of a Hebrew slave girl from her master that is at the time of being a na'arah, so too here [she is] a na'arah." [In other words: the similarity is the timing.] Problem: The sugya does not mean at all that it needs to learn that it is dealing with the day she is a na'arah but rather about the question "From where do we know that the money belongs to the father?"! Solution: "A similar going out"—i.e. under the authority of her master, if there was money there, then he could have given it to the master in order that she go out from him; so too here where there is money in her going out of the authority of the father, since it is her father, and this is also the meaning of the question. [In other words: the similarity is the existence of money.]

The resolution of how we know that the father receives the money goes back to the original source—the fact that when the daughter sold into slavery goes free, the master does not receive a payment. Had there been a payment, the master would have received it. Thus, logically, in the case of betrothal, where there is a payment, the money goes to the father, who is akin to the master.