If you could clarify your post as obviously the scenario you have presented makes sense.
I would assume there has to be some sort of agreement that was made when the daughter was added to the account. People cannot be just added at random. Often in the legal mumbo jumbo that is explained (or mailed out that nobody ever reads) you find out what the responsibility of the “signer” is. I would bet that it was disclosed that the daughter was responsible but it was not clear and/or she forgot.
You don’t have to actually “sign” to get the card and be held responsible as long as the details were disclosed. Companies are going to disclose and as Jae knows that if they don’t, they get into a lot of legal trouble.
Actually, people CAN just be added “at random”. That’s why they aren’t liable for the charges. Call any of your current credit cards and attempt to add your pet as an authorized user. Just make sure you give them a middle name 😉