Never in 100+ rentals, for me. Then again, I said the same thing about my AAA card at hotels until ONE TIME, so if you don't have a card, use it at your own risk.
I have not but I use my Costco Visa and I can't recall if they looked at the back or not. That said as Covid hit last year we wanted to get our son out of CA due to the impending closure, but at his age rental cars were not an option... until we looked into the Costco membership. He got a Costco membership and was able to rent a car, however there were issues picking it up and he had to go to the airport (long story) but through all of it they did verify he was a Costco member and both rental car locations. The only reason I know is we were on the phone trying to help him through the whole ordeal. Unsure if they were checking due to his age or not.
Not once since 2013, the first time we utilized Costco. While I have never been asked for my membership, I’ve given them my Costco credit card, which has my membership number, every time.