A couple of answers.
I was only ID-ed in nudist resorts where I rented a room or apartment from the resort proper (Koversada, Playa Vera). In Cap or Costa Natura we rented from the owner and didn't show any ID. And surely none at the entrance. In German spas you just buy a ticket and you can do it in cash. I took the regular passes in Sinsheim and Friedrich/Caracalla (which give me 15% discount) and I did provide my name and address for this (but I suspect I could have provided fake data).
Of course, everything I do can be tracked down to me by police but I don't like to be policed by private entities.
Some German spas have an over 16 rule. Some don't. Not sure if this is to not expose children to nudity (or the other way round), I suspect it is more because children are noisy. In Friedrichsbad the rule is clearly no under 16 as it is supposed to be a relaxing place - you can't even have an orange juice there. In Sinsheim 80% is textile but there too only children under 3 yo and adults over 16 are allowed. In the nudist area, nobody under 16.