Since mother offered to share, I don't think it would have been a problem to accept. I don't think it was crossing boundaries on the child's food. This wasn't her meal. Her meal was coming. If you helped yourself without the offer, that would have been rude. Since she offered, yum! (I wouldn't help myself to a second one, though)
Is this only because she's a minor? I don't think every appetizer is automatically "for the table" unless something is actually said. There's no way I'd feel entitled to a friend's appetizer if it looked like she was ordering it as a complement to her meal rather than as a shared dish, and that would extend to a friend's kid too.
At what point did I imply the app was "for the table" or that the OP or anyone else was "entitled?" The mother offered
. I don't see this as rude, or the OP accepting a cheese stick as rude either. She declined. Totally fine. If the OP or other table guest just dived in on someone else's app, that is an entirely different story.
My family generally order appetizers to share. If we're working on separate tickets, one group buys one, the other group buys one, and it's a free-for-all after that.
I have no idea what the typical norm is for this family or what their thoughts are on sharing an appetizer.
I think it's polite to offer, not necessary, not rude to not
offer. In this case, mom offered. Agreed that the other table guest could have ordered appetizers too if they really wanted to and could have declined the offer.
Share an app, not share an app, totally circumstantial.
I didn't even consider the soup/salad situation, but usually those are pretty automatic, are they not? Everybody grab a fork, dig in!