we use Stripe Element in our service in Australia. By defaults the ZIP is not shown - we see only Card Number, MM/YY and CVC. If I use a US registered card, such as the test number, then the ZIP field is added automatically (although it is displayed as Postcode here, not zip).
We have registrations from around the world and never had an issue with it.
Thanks Russell for sharing your experience. This is something handled by Stripe (we just use their api for processing payment), so if you want to know the details, better contact Stripe support and ask them about it (they provide good support)