I doubt it has anything to do with verifying age though. More likely they know people are hesitant to sign up and pay for youtube, but if they can get your credit card in other ways, then later it easier to get you to pay for something because then you just have to press "ok" since your details are already entered.
Plus this gives them your identity, which is relevant because their business model nowadays is based around tracking people.
Plus this gives them your identity, which is relevant because their business model nowadays is based around tracking people.