Fun fact: DuckDuckGo has this domain because Google gave it to them after acquiring it for unrelated reasons. It had been the domain of video codec developer Duck Corporation, which changed its name to On2 and was later acquired by Google primarily to get its patent portfolio for the WebM effort.
Google wouldn't assign any IP over without some terms and conditions, which may be what's limiting them from adopting it.
They should drop the "DuckGo". Just "Duck". It's cleaner.