“I want to know why it matters whether the child with you is your offspring, and not another child that just happens to be with you. They are asking for proof that this is your child, and I’m asking how does that matter according to the law?”
It would matter according to the law because, last I heard anyway, in Ohio a parent can provide an alcoholic beverage to their minor child, even in a bar or restaurant (something not everyone knows about).
So, if the child with you is your offspring, you can legally provide the child with an alcoholic beverage anyhow, so it doesn’t matter because they can’t do anything about it anyway.
If the child with you is not your offspring, you can not legally provide the child with an alcoholic beverage, some adults have been buying alcoholic beverages to provide to minors, they presume that’s what you are trying to do, and they want to and think they can do something about that.
Wow, that’s pretty convoluted.