Cash App is a debit card. Apple Cash is a prepaid debit cards. Apple Cash card by default is in your Apple Wallet. Cash App card may be added to Apple Wallet for use with Apple Pay and to transfer funds back and forth between Apple Cash. The accounts are not linked.
If you want to use Cash App to complete transactions with the Apple App Store it must be added as a payment method in your Apple Account. Please review the Apple Support article on adding payment methods to Apple Account.
Add a payment method to your Apple Account - Apple Support