How are all atoms of the same element the same?
All atoms are made of three constituent particles, electrons, protons and neutrons. Electrons are negatively charged particles, protons have a positive charge of equal magnitude and neutrons are electrically neutral. Atoms are inherently neutral particles; this is due to the fact that the number of electrons and protons is the same.
It is the number of protons (and consequently the number of electrons) in an atom that is unique for all elements of a particular element. Atoms of different elements only have to differ in the number of protons they have. The number of protons also provides the atomic number of an element.
Another property of atoms, the atomic mass is primarily due to the number of protons and neutrons in an atom as electrons have a negligible mass. Atoms of the same element can differ in their atomic mass if the number of neutrons is different. These are called isotopes of an element, i.e. forms with the same atomic number but different atomic mass.