The most characteristic property of an element is its atomic number. The atomic number of an element represents the number of protons in the nucleus of its atoms. Each element on the periodic table has a unique number of protons, which determines its identity.

Option 1, density, can vary for different elements, and it is not a specific characteristic that can uniquely identify an element.

Option 2, boiling point, also varies for different elements and is influenced by various factors such as molecular structure and intermolecular forces.

Option 3, mass number, represents the total number of protons and neutrons in an atom. While it can be used to identify isotopes of an element, it does not solely define the element.

Therefore, the correct answer is option 4, atomic number, as it is the most characteristic property that distinguishes one element from another.