Suppose we are working in an "arrows-only" definition of a category such as given in Mac Lane's "Categories of the Working Mathematician" (1998) p.279 or on nlab. How can we formulate the definition of a product in such a category? I do not see how to do this without implicitly referring to objects (identity arrows).
Using the notation of nlab, the following is a fibered product: if $x, y$ are arrows with $t(x) = t(y)$, then their fiber product is the pair of arrows $u, v$ with $t(u) = s(x)$, $t(v) = s(y)$, and $s(u) = s(v)$ such that for any pair of arrows $a, b$ with $s(a) = s(b)$, $t(a) = t(u) \, (= s(x))$ and $t(b) = t(v) \, (= s(y))$ and such that $x \circ a = y \circ b$, there is a unique arrow $c$ having $s(c) = s(a) = s(b)$, $t(c) = s(u) = s(v)$, and $a = u \circ c$, $b = v \circ c$.
To define a plain product, suppose the category has a final object (that is, of course, that there exists an arrow $f$ such that for any arrow $x$ there exists a unique arrow $x'$ with $s(x') = s(x)$ and $t(x') = f$) and replace $x$ and $y$ by $x'$ and $y'$ in the above. If it doesn't have one, of course you can just add one.
Can't get any farther away from identity arrows than that; you need to be able to specify sources and targets to define composition.
Exactly the same as in the objects-and-arrows presentation of category!
Working with the “arrows-only” definition of a category doesn't mean you can't talk about objects, it just means that they're themselves a defined notion. Some constructions can be very nicely given in purely arrows-only language, but for many things — and I think product is one — it seems most natural to define “objects” and then to use them in the statements of further definitions.
You can certainly then unfold the definition to give it in a way that doesn't mention objects. But I think it's important to note that you don't need to do this, and a priori, no obvious big reasons one would want to!
Edit: Reading Martin B's comment, I realise I may well have misunderstood the intent of your question. I'm leaving this answer, though, as I think it's still a point worth making!