Hi,
You have two characteristics. These allow you to generate variations of your main product, one variant per combination of the values of both characteristics.
So it's not about adding both selected values in the cart, but rather about adding to the cart the currently selected variant.
Now I suppose that you set names which are wrong in your variants, and thus you think that you add only "one value" but that's not possible as it's the whole variant which is added to the cart.
So fro example, if you select "Snap Button Bracelet - Blue" and then "Jamaica" and then click on the add to cart button, you're adding the variant for both of these values. And what you see in the cart is the product name that you set in that variant.
Also, it's similar for the image being displayed. If you want to display both a flag and an image of a strap, you need to add both images in each variants, since each variant is a combination of both values selected for the variant.
I hope this will help you understand better the situation and what you need to do.