The answer is right, and we can make this precise. As this answer says as well, there is the Elementary Theory of the Category of Sets (ETCS), which is in the language of category theory, and there is ZFC, the well-known axiomatisation of set theory. ZFC is really stronger than ETCS, the answer I linked goes in more detail, but point is that we can add axioms (a variant of replacement) to ETCS to make it in fact equiconsistent with ZFC.
Edit. From a comment from Noah Schweber: a survey of possibly-relevant weaker fragments of ZFC can be found here. I thought this might be relevant to point out here as well.
Edit 2. Upon reading the Quora answer again I should say that I agree with the technical part of the answer, namely that set theory and category theory can encode the same things. That was the point of my answer here. I do not agree that everything in mathematics is a "specialisation" of (naive) set theory. For example, I highly doubt many mathematicians view the natural numbers as the von Neumann ordinals.