This is why you have a relay in the middle.

Then, Chase knows you've verified your ID somewhere, the relay knows that some Chase user verified themselves at Pornhub, and Pornhub knows that the user is over 18, without knowing their identity or what bank they're using.

You could also do this with ZKPs and device integrity protection. The latter is more secure but more complex, the former is much simpler and openness friendly.

I think the key here is “a trusted third party” more than “bank”.

Also, I’d rather a company know “he has an account at Bank of America” than “His full government name is Bit Masher and his driving license number is 9”

Well, they already know your credit card issuer, it's not that wild. Really Visa/MasterCard should offer age verification on their network... They have all the necessary components.