Category theory is generally not used in a foundational way. That it can be, doesn't make every description of a categorical concept adequate for that purpose. The vast majority of presentations of category theory are not foundational descriptions and thus assume some pre-existing foundations, usually some variation of set theory. Type theory is another foundation that is often used (and also has close connections to category theory).
Your proposed approach just doesn't make sense. It does not make sense to talk about the product of objects from different categories.
Most "categorical" foundations are actually formulations of the category of sets, e.g. ETCS, the elementary theory of the category of sets. More generally, topos theory provides a kind of generalized notion of "category of set-like things" and has a lot of relevance to foundational concerns. Even from the perspective of category theory, it doesn't really make sense to start at the "category" of categories. Why not start at the 4-category of 3-categories instead? Too avoid going to far afield, my point is simply that the "category" of categories is not the natural base case.
Nevertheless, there have been attempts to directly axiomatize a "category" of categories or a 2-category of categories. The way to handle products in these cases is simple: you just take the universal property of products as an axiom. (Indeed, this is also exactly how products of sets are formalized in ETCS.) That is, we axiomatically assert that given two categories $\mathbb{C}_1$ and $\mathbb{C}_2$, there exists another category which we'll call $\mathbb{C}_1\times\mathbb{C}_2$ such that there are two functors $\pi_i : \mathbb{C}_1\times\mathbb{C}_2\to\mathbb{C}_i$, and for any category $\mathbb{D}$ and pair of functors $f_1 : \mathbb{D}\to\mathbb{C}_1$ and $f_2 : \mathbb{D}\to\mathbb{C}_2$ there is a functor $\langle f_1, f_2 \rangle : \mathbb{D}\to\mathbb{C}_1\times\mathbb{C}_2$ such that $\langle \pi_1, \pi_2 \rangle = id$ and $\pi_i \circ \langle f_1, f_2 \rangle = f_i$. "Category" and "functor" would presumably be primitive concepts in such an axiomatization. If the axiomatization otherwise did an adequate job of formalizing the notion of "category" and "functor", we'd be able to prove that the usual set-theoretic definition of product category is a model of the axiomatically characterized product category. In an axiomatization like the above, objects of the categories are likely to be identified with functors $\mathbf{1}\to\mathbb{C}$, from which we can already start to see that an "object" (in this sense) of $\mathbb{C}_1\times\mathbb{C}_2$, i.e. a functor $X : \mathbf{1}\to\mathbb{C}_1\times\mathbb{C}_2$, gives rise to a pair of "objects", $\pi_i \circ X : \mathbf{1}\to\mathbb{C}_i$ and vice versa.
