Once you get to the second cozy cabin at rank four, you can change your island's appearance, but it's minimal. You can go to your main cabin and choose Change Hideaway's Appearance. You'll need an island prism which you can buy off the market board, or you can craft yourself with any eight crafters from the mainland game. They are located under Master Recipes Nine at the bottom of the screen, so you can change and adjust to your heart's content.