Hey Thomas,
The lookup for country is a custom field. You are likely not seeing the values in the form itself, because the marketing service user does not have access to the entity (crm user do, which is why you can see it in the form designer).
The marketing service user security role should have full access for this entity:
(also relevance search needs to turned on for that entity)
However, I wouldn't recommend on using this country field in your customer facings forms. We've used it for a while and it wasn't ideal for our conversions. The fact that we can use lookup fields is very cool, but it is just not very intuitive for users, it sometimes takes a while to load (makes users think it is not working) and it crashes the form if people are using autofill options in browsers. If I autofill "Deutschland" in my browser, I will get an error saying 'enter a valid country', but users don't know this has to do with the fact that you typing "deutschland' does not associate the GUID of that record.
So in the end we stopped using this lookup. We still wanted to capture countries however, which is why we went for a country option set in the form. This is very common in webdesign, works fast in all browsers and is intuitive.
The lookup is still the preferred option in CRM which is why we match the submitted option set value to the country record in our country entity. So we still end up with a valid country lookup in the lead/contact records, but we don't annoy our users with the badly performing lookup form on our website. Our option set has the same ISO code country names as the custom country entity and we match this to the contacts/leads with 2 seperate flows:
