I've never seen one of those gift cards in plastic forms. This seems indeed to be very american ;) Every gift card I ever got or bought was just a piece of paper. And my only store loyalty cards are actually from the US (although they exist here as well).
Here they are much more commonly plastic. They look like a credit card and are on display at stores. It is just a much nicer presentation for a gift than a piece of paper you printed. It's also more convenient to hang onto. It doesn't have to be folded and get beaten up in your wallet when you don't use it for 2 months.
The only time you have something printed on paper is if you got it off a credit card rewards site for the most part.
