“Brand registry” means that you have registered your brand on your trademark register, whereas “brand authorization” means that you have sold your products using someone else’s brand name. If you want to build your brand on Amazon, then the Amazon brand registry will be best for you because when you sell your products with your brand name, then your brand will be ranked on Amazon. But when you use someone else’s brand name to sell your products, that brand will be built, not yours. Whenever the authorised brand owner wants, he will sue you, and you will not be able to sell that product. So, I will suggest that you go with the brand registry if you want to build your brand.