Question

How does the atomic structure relate to element identity?

Answer 1

number of protons = atomic number = element

Answer 2

does that answer your question?

Answer 3

most chemistry is defined by the outer orbital of electrons. and the relative attraction of the psoitive nucleus.
let us use sodium and chloride as examples. the electrons are at the same orbital, same "distance" from the nucleus as they are in the same line. sodium however has a smaller positive charge than chloride (its atomic number is lower, less protons. less charge. therefore electrons are bound not as strong as electrons of a chloride nucleus. therefore sodium will "tend" to lose ions when in proximity to chloride (and in a solution but let us put it aside) becoming an cation. chloride tends to scavenge electrons as relatively its positive "pull is stronger. thus becoming an anion.
furthermore. the tendency to lose ions or attract them is actually acidity (and alkalineity. i dont remember formally which is which . i am far from this material, but u get the idea.

these relative pulls over electrons neednt end up in losing an electron altoghther. covalent bonds in wehich electrons are shared are sometimes asymmetric , as the couple of electrons "spends" more time" around the nucleus with the greater pull (right of the periodic table.
oxidation is the process in which oxygen (second to the right in its row) "steals electrons from , lets say, metals (left of the table, weaker "pull" - thus causing rust.
this is what i can come up with at the moment. hope it helped