The word branded can mean "associated with a certain brand or trademark." So, a branded greeting would be a greeting that identifies the company name within the greeting itself, or maybe includes the words of a company slogan. These would be examples of branded greetings:
Hello, this is Coca-Cola. May I help you?
Hello. Live better with Wal-Mart. How can I assist you today?
Your company is asking its technicians to answer support calls with an unbranded greeting:
Hello, this is customer support. How may I assist you?
As for why a company would prefer to use unbranded greetings, I suppose there are a number of possible reasons. Perhaps management thinks it would tacky and commercialized to explicitly state the name of a brand, therefore sounding less professional. Perhaps, because it's a customer service line, many of the clients are having trouble when they call, and they don't want to overuse the brand name during troubleshooting, out of fear of associating the brand name with a negative experience. Or maybe the customer support function is outsourced to a third party, but the main corporations doesn't want its customers to know that, say, Acme Company's customer support is being handled by specialists from the Pinnacle Corporation.