I think many children these days are not taught the propriety of greetings and things like shaking hands. My sister and I were taught as children that if someone extends their hand to shake yours, you shake hands back. We both taught our children the same way.
Maybe what you see as a peeve of hers is really her way of trying to teach girls the importance of a proper handshake. It sounds like a subject you haven't brought up with your daughter and that's part of what is making her feel awkward about it. Nothing wrong on your part, I didn't mean it like that. I'm just saying that it being something new to her doesn't necessarily make it bad or wrong. You can take the lead and teach her about it on your own.
It IS important, in my opinion, for even girls to know the importance of a confident and firm hand shake.
Maybe if your daughter practices with you and shakes the woman's hand nicely and firmly next time, the woman will feel her mission in life has been accomplished. :)
Kids don't have to offer to shake someone's hand, but if a hand is offered, I think they should know how to properly respond, that's all I'm saying.