I'm torn on this. If the mother had offered up part of the daughter's entree (e.g. french fries), I would absolutely agree that this was not ok. For an appetizer it's not so clear-cut to me for several reasons.
1) Like some PPs, I have always seen appetizers as something to be shared. If that's the norm in this family, then the mother may have assumed that the daughter knew the appetizer would be shared. Often I've seen someone order an appetizer and offer it to the whole table once it came out. Sometimes I've seen it shared within a particular subgroup (e.g., within a family or among a few people planning to split the cost). In the latter case, it usually was a collective decision by the subgroup. However, I don't recall having ever dined with someone who ordered an appetizer to consume completely by themselves, unless they were ordering it in lieu of an entree. I've seen people order, e.g., an appetizer and a salad to serve as their meal. They usually specified that they were ordering the appetizer as their meal and had it brought out with the other entrees. In the OP's case, it was apparently ordered like a normal appetizer.
2) I wonder whether the daughter was actually "authorized" to order an appetizer for her personal consumption. If an appetizer (especially a personal appetizer) is not something the daughter has blanket permission to order, then the mother may have decided to avoid publicly contradicting the order and instead treat it as if Daughter was ordering the appetizer on behalf of their family (making it mom's prerogative as parent and payer-of-the-bill to offer it around). When I was a kid, meals out were generally an entree and a drink (soda, water, milk, etc.) for each person. Occasionally an appetizer (for the entire table) or dessert, at the discretion of whoever was paying. My parents never really restricted what we kids could order as our entrees--we could decide whether we wanted a large entree or a small one; we were welcome to add on the salad bar, etc. But I would never have ordered an appetizer or a dessert without discussing it with them first. I still wouldn't if the meal was their treat. IME, those simply aren't part of a meal out unless the person paying the bill specifically offers it.
I think it depends very much on the family rules/customs. If the daughter is typically allowed to order an appetizer for her own personal consumption as part of her meal, then I'd say it's rude to change the rules on her with no warning and start sharing her food around the table. But if that's not an established custom in their family, then I lean towards the parent paying the bill having a perfect right to insist on sharing the appetizer.