According to wiki reference here:
In its formal representation, the law of identity is written "a = a" or "For all x: x = x", where a or x refer to a term rather than a proposition, and thus the law of identity is not used in propositional logic. It is that which is expressed by the equals sign "=", the notion of identity or equality. It can also be written less formally as A is A.
In logical discourse, violations of the law of identity result in the informal logical fallacy known as equivocation. That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings without introducing ambiguity into the discourse – even though the different meanings are conventionally prescribed to that term.
So if law of identity doesn't exist, there'll be no Equivocation fallacy any more and we can say more equivocally confused things and we're still considered logically valid, but cannot be sound since our macro empirical world certainly doesn't look like this by any measure, even the same word clearly corresponds to different things under different contexts.
However, in our quantum world this law may not exist according to the same wiki reference above:
Schrödinger logics are logical systems in which the principle of identity is not true in general. The intuitive motivation for these logics is both Erwin Schrödinger's thesis (which has been advanced by other authors) that identity lacks sense for elementary particles of modern physics, and the way which physicists deal with this concept; normally, they understand identity as meaning indistinguishability (agreement with respect to attributes).
So in this imagined quantum world with really indistinguishable objects, you know there're many objects but you cannot have a predetermined clear way to choose a specific object, not even based on their relative space position as if still can be easily meshed and have a choice function to pick one out from our macro experience.