The point is your personal FB with your full name, work & family details potentially becomes accessible to her.
To be honest I can't see how, my punting FB is 'open' and I don't get suggestions for people I actually know, my personal FB is 'closed' in that I mean it limits, as far as possible, who can see what or send friend requests. I do get suggestions for people I know on my personal one but I presume that comes via friends or people I contact.
I've never been to or had any contact with anyone in Mexico on my punting FB yet I have suggestions for people I might know who live there. It's the same with the USA, I've only been to the north but get suggestions for people in the south. Likewise various cities throughout the UK.
The point I'm making, up to now, my 2 FB accounts have never had the same suggestions or suggested each other, nor have they linked in any way.
I'm no expert in social media but from what you say as I use the same computer and IP address for both FB accounts they should link in some way or at least suggest each other as someone you may know, they don't though.
On my personal FB I get more suggestions for people in my local area and on my punting one more from outside my local area. Like I said I must just be lucky.