No, altough the wording might have been misleading—I was refering to the second paragraph, while you seem to have applied it to the first. I hope this clarifies the situation a bit:

For any category, the product A * B of two objects A, B (if it exists) is unique only up to isomorphism (this follows from the universal mapping property).

In general, products A * B and C * D need not be isomorphic (and seldomly are), and neither do products from different categories. Many of the commonly-used products actually arise from a product in a suitable category, the cartesian product, for example, is a product in the category of sets.