Menu/Module display output hierarchy

When you display a category listing menu. Whether it's displaying the categories of the top level or their sub categories, it will display the same. That's normal.

The option "Synchronize with currently displayed item when possible" means that when you are on a category/products listing menu which is displaying the elements of a category (that's always the case) the module would also display its elements with that main category for reference.

Take the categories module on the bottom left of the demo website: www.demo.hikashop.com/index.php?lang=en
When you click on the "Shoes" category, the page will display the products listing of the "Shoes" category. But the categories modules still display the same categories. But if you turn that option on for that module, on that page, it will try to display the sub categories of the "Shoes" category and since there are no such sub categories, it won't display anything.
"When possible" means that when the page is not a category/product listing, it's using the "associated category" as "reference" category for the elements. That's why on the home page, you see these categories which are the sub categories of the main product category of the shop.

You can't remove the associated category. That's also normal. If ther eis no reference category for a listing, the system won't know what products/categories to display.