When I started using them, they did this by checking against Paypal, with whom (admittedly to my regret) I had already verified myself. I wasn't asked to provide a copy of my ID to them directly, at least, or to provide it anew to one of those random ID verification companies that are popping up out of the woodwork.
It also just bothers me less in this case than in most because, no matter who you buy from, if you ever need to verify ownership of your account/domains, you may eventually be asked to show ID/verify your identity anyway, and if you can't prove you're the person who bought the domain then you risk losing it (say, by not being able to regain control of it after it is stolen). And if it's a domain you've tied your email or business to, and you've pre-payed 10 years, that would suck majorly.
So I feel about it more or less how I feel about my bank needing ID, personally. But I definitely get why others may not agree/may have a different use case to begin with.
I think also there is a big problem with scammers using stolen credit cards to buy domains, which they use to send phishing email or operate malicious websites. Preventing this at least makes way more sense as a motive than "protect the children by identifying all of them".
If you buy from elsewhere, you can find a way to avoid the ID verification, but most places will only take digital payment, so they still probably end up with your card number and name.
I'm not a fan at all of age verification laws and websites requiring ID, but this one I tolerate, personally. But I won't blame anyone for not doing the same.
