Yes and no. Sometimes when you buy a brand name shoe you are paying for the reputation of a brand that has reliable and quality products. When you buy running shoes from Adidas, for example, you are probably betting on the long-standing history and success the company has that can in some ways help you predict what the quality of their shoes will be like.
Name brands aren’t for everyone though, and if you can find a pair of generic or lesser-known shoes that work well for you then more power to you.