Avoid ice crystals in frappuccino
There are two main factors here:


As captured in comments, a handheld stick blender just isn't powerful enough.  It has nowhere near the same power as the sort of larger commercial blender used by big chains like Starbucks.  Not only that, but it can't create the same sort of turbulence as an enclosed blender, because you're immersing it into an open container - if it did, everything would just slop over the sides.  Makes a sad frappuccino.  That turbulence and power makes for very, very fine ice crystals which stay suspended in the mix.
There are indeed additives.  Starbucks' commercial syrup bases include a number of emulsifiers (notably xanthan gum and potassium sorbate: see this similar answer) which thicken the mixture and keep everything nicely integrated for a smooth texture.  A banana also contributes thickening power and its own natural emulsifiers.  If you want the same effect without the banana flavor, you could try adding egg (bananas can actually serve as an egg substitute, and eggs and are frequently used as an emulsifier in other drinks) or your own xanthan gum, etc.  Such emulsifiers aren't common at groceries but can be easily found from modernist cooking suppliers.  I believe there are quite a few resources and related questions on this very site.


If you have access to a full-fledged blender, and have just been using the handheld model for convenience, then that's the easiest swap to make.  Otherwise, aim for emulsifiers - they'll probably be a substantial improvement even without an equipment change, although you'll likely have to blend for longer than you have been otherwise.