The stored card details are linked to your email address, if you type in your email address in the pop up field, Stripe will send you the verification code to your mobile phone.  If you ignore the code when it changes to the screen "Enter your verification code",  in the top left corner is a back arrow, go back a screen to where you entered your email.  Your email will already be populated but you can then type new card details and then process the entry with the changed card details.

Stripe will alway remember the last set of card details you used if you have asked it to remember them.  For further information about Stripe Checkout Details visit https://stripe.com/checkout/info