Does using something shared between siblings or the parents' personal belongings like a Mushaf or a book contradict the Hadith: "And a twig of Arak"?

There is no harm in using each other's belongings that are customarily used without explicit permission, because customary permission is equivalent to explicit permission. The hadith "Whoever usurps the right of a Muslim person by his oath..." refers to taking money without the owner's consent, and it does not contradict the situation mentioned as long as the items are taken with the owner's consent. However, if it is known that the owners are not pleased, then it is not permissible to take their belongings.