We do very few dives through shops, usually just tank rentals and shore dives. Last trip the shop was happy to see any card to rent us tanks. Sometimes we get asked and sometimes we don't.
We did choose to do a boat dive. We were shore diving just down the street from the shop that offered the dive we wanted to do. During our SI we stopped by the shop to set up the trip, still half dressed in our wetsuits. They asked about out certs and "last dive", looking at my watch said 15 minutes ago. They never even asked for cards.
I have never been asked for a log book, again limited in the type of diving we do. In theory I understand, but also it is something so easy to fake it seems pointless to me.