Ha! I have a similar story.

I tried to buy a mobile phone over the internet.

After processing my details in their secure website, and trying to pay by paypal, they had the temerity to -email- two days later and ask for a copy of my passport or drivers licence.

They suggested to send it either by fax or by email.

I was absolutely horrified. There was no way -anyone- on the internet was getting a photocopy of my passport, essentially my identity.

As in your case, the woman didn't see email as anything less than utterly secure.

Nor did she see a problem with asking for it by email, when I had included two phone numbers as well.