In the case of Bibles--not just any other book, but Bibles--especially family Bibles such as many Southern families keep and pass down through the generations, it is common to record all sorts of important events in the lives and histories of the family members as they happen. The federal government even accepts such records as proof of age or birth in lieu of a birth certificate or census record. For someone to ask the president of the United States to sign their copy of the Bible is nothing unusual.
And as for the signer, President Trump, politicians as a courtesy routinely sign books or hats or shirts or pennants or print programs and other items when people in crowds hold such items out to them for signing, just like athletes do. So there is nothing remarkable there, either.