Okay I found the solution. A view in the mySQL table helped to understand how the Event Organiser plugin is using WordPress. The Event Categories a just normal categories with the taxonomy event-category. The only function needed is get_term_by().
// Get term by slug ''test-cat'' in event-category taxonomy.
$cat = get_term_by('slug', $eo_event_loop->query['event-category'], 'event-category');
echo $cat->name; // Output: Test Category