Choosing products for a registered brand

Newbie question: If a seller successfully goes through the process of registering a brand, can that seller purchase some goods on Alibaba (or any supplier) and brand the packaging or even the product with the registered brand name/logo and then sell that as a unique product on Amazon FBA if other registered brands have already done the same? Or would the seller need to find a supplier whose product isn’t listed yet? I’m thinking simple household and kitchen goods like glassware – no chemicals, electronics, etc for the US market.