I'm self-learning category theory from multiple sources. In Bartosz Milewski's lectures, he introduces the categorical product, via the universal construction. He then shows that for several useful categories, the product is defined for every pair of objects, and often resembles a Cartesian product.
On the other hand, in Coecke and Paquette's Categories for the practicing physicist, they introduce a symmetric monoidal category as a category that simply comes equipped with a bifunctor $\otimes$ and identity object $I$ that satisfy certain relations. This $\otimes$ operator also seems to be similar to (and sometimes the same as) a Cartesian product.
My question is, are these two concepts actually the same? By which I suppose I mean, is it the case that if a categorical product exists for every pair of objects, does it necessarily always give the structure of a (symmetric?) monoidal category, and conversely, is the $\otimes$ operator in a symmetric monoidal category always given by the categorical product?
More broadly, regardless of the answers to the above questions, how should I think about the relationship between these two ways of defining Cartesian product like operations in category theory?