Here, Andreas Blass wrote:
The axiom of regularity is clearly true with this understanding of what a set is. It expresses the idea that the stages of the cumulative hierarchy come in a well-ordered sequence.
One can easily convince oneself that this is true, and the main point here is that the usual ordering on the class of ordinals is a well-founded relation. This wikipedia article also states that a relation is well-founded if and only if it allows induction/recursion. Thus, from this point of view, it seems to me that the axiom of regularity is equivalent to the principle of transfinite recursion/induction.
But this contradicts the article http://jdh.hamkins.org/transfinite-recursion-as-a-fundamental-principle-in-set-theory/ by Hamkins which says that transfinite recursion is in fact the axiom of replacement.
How to resolve this seemingly paradoxical situation?
You are talking about two totally different "principles of transfinite recursion". The first one is the statement that you can do transfinite induction on the relation $\in$ on the universe. The second one is the statement that given any well-ordering $<$ on a set, you can do transfinite recursion using arbitrary class functions on that set. Both of these statements involve a form of recursion or induction, but other than that they have little in common.