No, you do not need a healthcare card for expat health insurance.
Whilst a healthcare card is a common feature of public insurances in Spain, a healthcare card is not provided for expat insurance as it is not required.
To use your insurance, you will act as a self paying or private patient, then pay upfront (or request a payment plan if the bill is very high), and submit all invoices/receipts through your Feather account to start the reimbursement process.
You can lean more about how to submit a claim here.
If you would like to learn more about expat health insurance or would like to sign up, you can do so here.