> Pretty much anything involved with money needs not just age but in most cases identity verification too.
"Pretty much anything", you say?
I've never had to provide proof of age, let alone identity to purchase things over the Internet. A credit card and related address were all that was required. (What weirdo jurisdiction do you live in where you have to provide ID to order from `newegg.com` or your local instance of a big chain grocery store?)
Hell, this was the case like fifteen, twenty years ago when I had subscriptions for several of the larger porn sites. "Give me your credit card number and assert that you're of legal age in your jurisdiction." were the sole pair of requirements.
> Online banking... online trading
This is a result of "anti-money-laundering" (aka "KYC") banking regulations and has nothing to do with being online transactions, rather than in-person. Long ago, it used to be that you could show up with a sufficiently large wad of cash and get a checking account opened, no question.
> ...online gambling...
Given the enormous set of online transactions I can do without providing proof of ID or age (like, suchas getting alcohol and tobacco delivered to my door) this is clearly the exceptional case, and the one that relevant politicos are using to act as a basis for the ongoing campaign of Erasing Sin from The Internet (and massively empowering authoritarians in the process). (The fact that the overlap between the politicos agitating for these regulations, and the politicos who are massively authoritarian is very, very large usually goes unremarked in the press.)