Is there a knock down argument showing that , in arithmetics, equality means identity?

I would personally make the following distinction: ${3+2}$ is an expression, meaning (from Wikipedia):

it is a finite combination of symbols that is well-formed according to rules that depend on context

In this context, one way to define the symbols ${3,2,+}$ would be via the Peano axioms (which you may already know about). The Peano axioms do indeed define ${3,2}$, and give us an abstract ruleset for deriving what is meant by ${3+2}$. In this case, the expression ${3+2}$, following that ruleset, evaluates to $5$.

I would say that

two and two makes four

sure is a finite combination of symbols, but the ruleset is not clear, and I would argue is not abstract enough to be considered a true Mathematical expression. You could pick up two apples, and then another two apples, and count four apples - but then who said the objects had to be apples? Why can't I pick up two pens, followed by two more pens? What do I even mean by a single pen? It's all too "real world" and is not abstract enough. We want Mathematics to really be a language that is more robust than everyday common spoken language, and that's why, after a certain level, we stop using phrases like "two and two makes four".

So I would argue that equality being viewed as identity is just a natural extension from thinking in real world terms and going to the abstract realm; to simplify the ruleset so we can unmistakably, unambiguously derive the result of an expression with no other possibility (although unfortunately, we cannot prove Mathematics doesn't have contradictions).

Let $f$ denote a function, which we consider as having one argument, possibly a tuple. Set theory reads $f(x)=y$ as $(x,\,y)\in f$, construing $f$ as a set of ordered tuples where if $(x,\,y),\,(x,\,z)\in f$ then $y=z$. The obvious asymmetry, that $f$ takes $x$ to $y$ while in general nothing takes $y$ uniquely to $x$, is encoded in the lack of an analogous rule that if $(x,\,y),\,(w,\,y)\in f$ then $w=x$. We might say the first argument in the tuple "makes" the second, if we want to relate this to familiar terminology.

We often consider $+$ a "binary operation", which Wikipedia describes as a "rule", rather than a function, to go from $x$ to $y$. But for any such rule, there is a function $f$ that gets the same values. The "active" aspect you refer to can be thought of as how one computes a function. (Formalizing this notion, and what can and can't be computed, is a branch of mathematics in its own right.) Indeed, computer programmers use "function" to mean something that does something, to compute or otherwise (if it doesn't return a computed value at all, it might be called a procedure).