I can't agree. If a stranger came up to me wanting to interrogate my kids (or kids I'm babysitting) about who I am and if they're with me willingly, I'd find it insulting, would worry it would upset the kids or make them nervous, wouldn't find it any her business, and would probably just find it downright creepy. You don't know her mental state or who you're giving permission to with these kinds of requests.
So to be clear, if there was stalking involved that is a different story.
If someone came up to my kids and started talking to them, I wouldn't care.
EVERY TIME I leave my house or go shopping I don't know the mental state of the people around me. Why let your paranoia get the better of you? I mean, is the US chalk FULL of child predators who come up, ask to speak to your children and then grab one and run away? How often does that happen in the continental US?
Is it REALLY worth your concern?
What could be the trade off if you engaged in a positive conversation with someone?
Would it upset the kids? I'd wager they would respond to it in ALMOST the exact same manner as you did. If you freaked out, guaranteed your kids would freak out. If you were unflappable, your kids would be more unflappable. Not to say it wouldn't warrant another conversation if things don't go smoothe. But how can you expect to be a Christian light if you aren't even willing to talk to a stranger in public?
Is this a cultural thing?
Would you think most Americans would react this way? Up here in Canada I've had numerous strangers come up to my kids in stores (my boy had angelic blonde curly hair that got a LOT of attention) and talk to them and it was totally fine. I've never had someone question whether they were my children or not BUT even if they did, I wouldn't let it bother me.