Christmas cards are used in Italy, but they are really out of fashion. My grandmother used them, and then followed a phone conversation with the closest persons after Christmas to chat about novelties. This was until the '80s.
After that, Christmas card started being used merely for commercial relationship, i.e. companies sending you a Christmas card just to sell you something or to announce their upcoming sales or the pre-sales (yes, that's a thing).
When I was an adult, Christmas card were used in the office by suppliers to 'thank you' for your business with them or just reming you they existed.
There are smaller cards that can be attached to gifts with the name of the recipient and/or a few festive words inside. These gifts are handed over in person, so no mailing.
I think it's always nice to receive physical mailing, but given the cost and the hardship of international mailing, there is really no point.
I used to send Christmas card to my clients, but this year I passed as I have lost contact with most of them anyway.