Why do a phone and a watch count as two different cards?
Because the credential is issued to the device rather than to you. Adding the same card to a phone and to a watch produces two credentials on one account, which is why they can appear separately on a statement, why a refund wants the device that paid, and why losing one does not affect the other.
The credential belongs to the device
What a phone presents is provisioned to that phone, not copied from the plastic, which the phone payment page states and which EMVCo's tokenisation work specifies.
Add the same card to a watch and the watch is provisioned in its own right. Two devices, two credentials, one account behind them, and no way to move one to the other.
This is not an implementation detail that leaks occasionally. It is the design, and almost every question people have about phones and watches at a terminal follows from it.
What that looks like on an account
Statements and applications often distinguish which device made a payment, because the credential identifies it. That is the property behind a familiar surprise: a card added to three devices looks like three cards in a wallet application, all pointing at one account.
Limits can also be applied per device where the rules allow it, since the verification available differs: a phone that checks a face and a watch that knows it is still on a wrist are not making the same statement about the holder.
None of that changes what the merchant sees, which is one credential and one transaction.
Refunds and the device that paid
A refund is normally applied to the credential that paid, so a shop that asks for the same device is not being awkward. Presenting the watch when the phone paid offers a different credential, and the terminal has no way to know the two belong together.
Where the shop can look up the original transaction another way, this stops mattering. At a counter with a card terminal and no receipt, it matters.
The practical rule is the boring one: pay and refund with the same device, and keep the receipt that identifies the transaction.
Losing one of them
Because each device has its own credential, one can be suspended without touching the other. A lost watch is a revoked watch credential, and the phone keeps working.
The account behind them is untouched by either revocation, which is the point: the plastic card is not cancelled because a device was lost, and the stolen phone page covers what to do and in which order.
The same independence applies when a device is replaced, since nothing is moved and everything is reissued, which the section's guide on changing phones takes up.
Frequently asked questions
Is it the same card on both devices?
One account, two credentials. Each device is provisioned in its own right, and neither can present what the other holds.
Why does my statement show the device?
Because the credential identifies it. That is also why wallet applications list the same card once per device.
Can I refund to a different device?
Often not at a terminal, because the refund follows the credential that paid. With the original transaction on record, a shop can usually do it anyway.
Does losing my watch cancel my card?
No. The watch credential is revoked and the card and the phone are unaffected, which is the reason credentials are issued per device.