I don't recall Anthropic's payment systems, but I use Paypal wherever supported. I don't think Paypal sends my address, but am not sure. I'm pretty sure they don't send the CC information.
And often, not even the name (e.g. have often had people use my CC to buy stuff (with my permission)).
Also, I still routinely buy stuff from one service that thinks I'm in a state I haven't lived in for over 20 years, because that's the address I provided back then.
So no, generally, sending your payment info doesn't equate to sending them my address.