In some categories there are more than one (isomorphic) direct products: For example in Set there are $A\times B$ and $B\times A$ products (as well as many others).
But only one of these products ($A\times B$) is considered "canonical". Is there any formal or informal term about the "main" of the products?
There is no point in singling out a specific product in general category theory, because all (non-evil) properties of a specific product hold for any product. In concrete examples this means you're free to choose the most convenient product for the task at hand.
Having additional structure like a partial order on the set of objects or morphisms doesn't change that fact, because the categorical product - as it is defined for general categories - simply doesn't "see" that structure (unless there is some deeper connection between the structure of the category and the given partial order). If you add structure to an existing algebraic object, you might wanna tinker the existing structure to respect the new one. E.g. there is the notion of a locally partially ordered category, which imposes the condition on morphisms to satisfy $f_1\le f_2 \wedge g_1\le g_2\implies g_1\circ f_1\le g_2\circ f_2$. But then again you won't find a general canonical choice of a product.