No, Strawberry Hill was sold in 1923 to St Mary’s University College which still owns the site. A recent exhibition of masterpieces from Horace Walpole's collection called
The Lost Treasures of Strawberry Hill was "sponsored by J. Stern & Co., whose Managing Partner, Jérôme Stern is a descendant of the French branch of the Stern banking dynasty." But that doesn't mean the family is still (co-)owner or manager of the mansion.