I'm not really sure that there is a question here;or if there is, I'm having trouble finding it; however:
Equality in mathematics has been more generally analysed in modern mathematics, that is using the formality of set theory, as an equivalence relation, this is a relation = on some set that satisfies the following axioms:
transitivity (if x=y & y=z then x=z)
reflexivity (it is always true that x=x for any x)
symmetry (if x=y then y=x)
These are easily seen to be satisfied by the usual notion of equality. The last two properties are so self-evident to Euclid that he doesn't mention them. But it is in fact useful to do so. The property that Euclid mentions is the following
1': euclidean ( if x=z and y=z then x=y)
and this together with either reflexivity or symmetry implies transitivity.
This idea has broadened into the idea of isomorphism in modern algebra with the work of Emmy Noether, and underlined by the Bourbaki school which took a strong line on this - one could term it Structuralism in mathematics, which has a broad sympathy with a parallel movement in the humanities also called Structuralism, prominent in linguistics (by Sassaure) and in Anthropology (by Levi-Strauss), but did not directly influence each other. In mathematics this current deepened into category theory, where one is not allowed to assert that two things are actually equal, but only that they are isomorphic. This is typified by what are known as universal properties which defines an object upto isomorphism. A good definition in category theory is one that is satisfied by some kind of universal property. That this add something fundamentally to mathematics, is seen by the fact, that multiplication (products) are dual to addition (coproduct), a discovery in basic mathematics (with the usual notion of basic) that had not been discerned before.
But there is more, two objects may not only be isomorphic, they may be isomorphic in more than one way, in which case we get higher order isomorphisms. This is the beginning of higher category theory. This may also be interpretable as proofs between types: for example, we may have two different proofs that two types are isomorphic, and then a proof that two proofs are in fact equivalent. This interpretation also allows ideas of homotopy theory in where we consider the 'shape' or 'topology' that all these proofs make (metaphorically, if one visualises a proof as a string of assertions, with one following another; the question is can we deform one string f proofs into another). This is the beginning of higher toposes.
In a different direction, Liebniz in philosophical investgation into indiscernables said that, uncontroversially, if two things are equal, then all predicates satisfied by the first must also be satisfied by the second; and controversially, if both objects satisfies the same set of predicates, then they must be the same object. In symbols:
for all x, y [x=y -> for all P ( Px <-> Py) ]
for all x, y [for all P ( Px <-> Py) -> x=y ]
This idea has discernable effects in physics which leads to Bose-Einstein statistics in which probability does not behave classically. For example, two photons with the same energy are not discernable which is unlike two identical coins which are in fact discernable.