First of all, you cannot override *PaymentExtensionInterface* - it is generated piece of code.

If you want to define a new field, similar to *agreement_ids* you should create *extension_attributes.xml* file in your Company/Module/etc directory with a content like this:

    <?xml version="1.0"?>
    <config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Api/etc/extension_attributes.xsd">
        <extension_attributes for="Magento\Quote\Api\Data\PaymentInterface">
            <attribute code="subscription_id" type="string" />
        </extension_attributes>
    </config>

Getters and setters for *subscribtion_id* attribute will be generated by Magento and will be present in *PaymentExtensionInterface*.
 
You can find more info here: https://devdocs.magento.com/guides/v2.3/howdoi/checkout/checkout_new_field.html

Unfortunately, example in docs is more about shipping than billing step, but take a look at Magento's *checkout-agreements* module for some tips.