Yes, but no. That's the whole point of the rebranding. You will not find the words "Good Old Games" anywhere in their official materials when referring to themselves. They are now "just" GOG.

They still use "good old game" as a tag for some of the games they sell. But they will never utter those words when referring to themselves in any capacity.

Is it silly? Maybe. But it's a valid marketing strategy. How effective is it? I don't know. Maybe ask BP.