Can I add multiple item colors to my cart without recreating the design?

Certainly! If you want multiple colors of the same product, once you click the “Add to Cart” button in the Customizer, click the “Add another” button on the right side of the page. This will duplicate the item in question while allowing you to choose another item color.
On the other hand, if you’re wanting the same design placed on a different product, there isn’t currently a function to do that. If you’re wanting a similar design on multiple products, the designs would simply need to be recreated/reuploaded for each different product.