Kleenex and Xerox are great examples of what I deem as true brands. When your nose is running you ask for a kleenex. Photocopies are frequently called xeroxes. Market dominance and consumer awareness worked together to turn a company name into a universal product identifier. Sometimes we call a tissue a tissue but frequently we call it a kleenex.
Not every brand has this distinct attribute of Kleenex and Xerox, but they all have strong recognition and loyal users. People request Coke A Cola over Pepsi or vice versa depending on personal preference. And there are local brands such as Maryland's Old Bay. It is the most popular seasoning for steaming crabs. People's home made crab seasoning is referred to as Old Bay. I bet you can name a few local brands from your area that have become more than a company product name. They all started out as product names and logos and grew in popularity to where the name and logo are valuable. Every business desires that phenomenon.
I wish we could revert to calling a logo a logo and recognizing the difference between product identity and what truly is a "brand." But I will be realistic. The term "Brand" has been used instead of logo for so long, it won't happen. I will advise you to aspire to have a brand, plan you're marketing and image efforts to work towards becoming a household name in your market.