Yes they do verify the other proof of identity (and they dont accept all of them), at least here in Canada. The data can serve for phishing though. If you think about it, the information stolen is very easy to get on anyone you barely know. I hope they do verify proof of identity in the states too.
Between, I deal with credit agencies almost everyday.
They don't here in the USA. Amazon will give out cards, and they have no way to verify you through any means but the honor policy really. I like the Canadian system much better. If there is any further verification it happens behind the scenes which would be scarier than not having it done at all really.
Are we talking about Amazon gift card or credit card? If credit card, are you sure they arent asking for identification number of verifiable source like drivers license or health insurance? It just dont seem realist to me that you can get a credit card by giving your name, adress and date of birth. But if your sure of that, fine, im glad that its at least verified here in Canada. I had people refused because of false identity. I also had the agency faxing back a demand because the identification number wasnt good and when you would check back with the customer youd notice that you had in fact, written a wrong number on the demand.