A coast or coastline is a geographical feature that exists independently of which state, country or other entity currently claims 'ownership' of it. For that reason, I would suggest that it should be treated as a singular entity and referred to as one coast or coastline.
For example, it would not be unusual to refer to the British coast or the British coastline in the singular (see Google search). If the entire British coast is regarded as a single entity, it follows that any contiguous sub-section of that must also be a single entity, irrespective of whether that contiguous section borders one country, or two or three countries (England, Scotland, Wales); or one county or several counties (counties are the main sub-divisions of England).
In practice, if the particular contiguous section of coast actually borders two or more areas with different names (as in OP's example), it is not unusual for the plural form to be used; it would be unlikely to be considered 'odd'; and it would be unlikely to be misunderstood. Nevertheless, I would argue that useage of the plural form in such cases would (strictly) be incorrect.