There needs to be a distinction made between the terms Peano axioms and Peano arithmetic. The OP is conflating the two terms, hence the confusion.
In this wikipedia article (as of 11/21/21), it says,
A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.
The math stackexchange post linked in the OP refers specifically to the theory of first-order Peano arithmetic where multiplication and addition operations are indeed postulated axiomatically. See this link and this link.
As for the original Peano axioms, we should refer to Arithmetices principia, nova methodo exposita by Giuseppe Peano, 1889. The two images below are from the archive here (around pages 23 and 29 of 49).
Later we see this.
A translation of Peano's work is given by From Frege to Gödel: A source book in mathematical logic by Jean van Heijenoort, 1967. We see that there are nine axioms (for natural numbers), and the multiplication is indeed given by a definition in the original work.
It needs to be stressed that there is no contradiction between the post that the OP linked and the historical fact. The post is talking about first-order Peano arithmetic; the Peano axioms refer to Peano's original list of axioms. It's just that Peano axioms are axioms for the theory of second-order Peano arithmetic as opposed to first-order Peano arithmetic. So both are correct.
To answer the questions directly, see below.
How does this square with the fact that Peano axioms are usually presented as a list of five (or nine) axioms that mainly talk about the successor operation (and the equality relation if we're not presupposing it) with no mention of addition or multiplication?
The list of Peano axioms is a list of five/nine axioms that only talk about the successor operation and no mention of addition/multiplication. The axioms for the system of first-order Peano arithmetic requires the induction axiom to be changed to a first-order axiom schema and additional axioms involving addition/multiplication added.
Was multiplication postulated as an axiom in Peano's original work, or was it given by a definition?
It was given by a definition.
Is the person in the post wrong?
No, because he's talking specifically about first-order Peano arithmetic.