No, it is not a signed Diary of a Young Girl book — that would be impossible. Apparently Anne had this Grimm’s Fairy Tales book and signed it or printed her name as a signal of her ownership, the book was not taken into hiding. By whatever route from there, the book landed in a used book shop in Amsterdam where “a couple” bought it. They did not realize the significance of the name until 1977: their children recognized the name. They wrote to Otto Frank, Anne’s father, who was the only surviving member of Anne’s immediate family. He replied by mail that they could keep it.
The book and Otto’s letter are both up for auction in May, by Swann Auction Galleries. I find nothing on their site about this.