I would treat them as totally different things. There is a Category of Sets, but a Set of Categories would be a bit harder to define. So axomatic set theory could be a specific case of category theory, I suppose. But you can probably do a Class of all categories. (A Class is sort of a set-theoretic way to get around Russel's paradox, incidentally, you usually use a Class to define categories, so...) Though that's actually quite an irrelevant point. It's a completely different language for describing mathematics. I think describing category theory as an alternative foundation for mathematics (you really mean topos theory here) is a bit of an exaggeration. it's technically true, but most mathematicians I know are using it as a powerful device to prove things in algebraic topology or geometry, etc.

Category theory and set theory are deeply interconnected, as both are foundational areas of mathematics but with different focuses. Set theory studies collections of elements and their relationships, serving as the groundwork for much of mathematics, while category theory abstracts and generalizes these ideas to focus on structures and their relationships.

One key connection is that the category of sets, called "Set," is a fundamental example in category theory. Its objects are sets, and its morphisms are functions between them. This shows how set theory can be seen as a special case of category theory. At the same time, many categories studied in category theory have underlying set structures. For example, groups, rings, or vector spaces are often built on sets with additional structure.

Another connection lies in how category theory generalizes set-theoretic ideas. Concepts like products, coproducts, limits, and colimits in category theory extend the familiar notions of Cartesian products or unions in set theory. The Yoneda Lemma, a cornerstone of category theory, relies heavily on set-theoretic intuition, as it connects abstract categorical concepts to concrete representations in terms of sets.

Topoi, a concept in category theory, also bridge the two fields. A topos generalizes set theory by providing a categorical framework that behaves like the category of sets but with additional logical structure. This allows set-theoretic reasoning to be carried out in a more abstract setting. For instance, the category of sets itself is the prototypical example of a topos.

However, there are also differences. Set theory often grapples with "size" issues, such as distinguishing between sets and proper classes. In category theory, similar concerns arise, especially with large categories, like the category of all sets, which cannot itself be a set. Set-theoretic tools are often used to handle these size issues by defining notions like small and large categories.

And there’s a philosophical aspect. Set theory forms the traditional foundation of mathematics with its ZFC axioms, focusing on elements and collections. Category theory offers an alternative foundation by emphasizing relationships and structures, often referred to as "structural mathematics." These perspectives aren’t in opposition but rather complementary, with each offering tools to understand mathematics from different angles.

In essence, category theory builds on and extends set theory while also relying on it for foundational concepts. The two fields are tightly intertwined, with set theory providing a concrete framework and category theory offering an abstract, flexible way to study and generalize mathematical structures.