Is it possible to add a role and aria-label to the div created on the popup calendar on the datepicker widget? 

I believe this is the correct division  <div id="ui-datepicker-div" ....  Something like this will work  <div id="ui-datepicker-div"  role="complementary" aria-label="JQuery Datepicker"  ...

When aria or HTML5 landmarks are used on the page, but not all content is included, an accessibility issue occurs.  

When using HTML5 or WAI-ARIA landmarks it is best practice to include all content on the page in landmarks. In this way users of assistive technologies can use the landmarks for navigating the page without losing track of content.

Thanks,
Russell