My LDS actually requires me to provide at least a pic of my buddy's c-card if I'm renting gear for someone they don't know. I honestly don't mind it, I'd rather work with a shop that is safety and liability minded.
I was once questioned about me renting gear for another diver......the shop confirmed with me that they were certified and then gave me everything I asked for. So, they asked but really that does nothing. Thankfully I am smarter than the friend in this case and would not provide an uncertified diver with a complete scuba kit.
One other time I was asked for my C-Card for tank fills. Once.