Comment by pimterry

The company doesn't actually keep your card details at all (at least, all reputable companies). They take the details to the payment processor at first purchase, but they then get swapped for a token which can be used to process transactions (usable only for transactions to you by this one vendor, so tokens can't be stolen/leaked, unlike card details) and then future transactions all just use the token.

When your card details change, all issued tokens generally stay valid, they're effectively independent. A payment card is basically an initial authentication process for the account, it's not really the payment method.